Grammik Algebra' Iusers.math.uoc.gr/~chrisk/LinAlg1-Notes.pdf · 2019. 5. 18. · Oloc' o mhdenoq¸roc tou A iI onom zetai idioq¸roc tou Agia thn idiotim i. TonÐzoume oti to mhdenikì

Shmei¸seic maj matoc MEM 106

Grammik 'Algebra I

Qr stoc Kourouni¸thc

TMHMA MAJHMATIKWN KAI EFARMOSMENWN

MAJHMATIKWN

PANEPISTHMIO KRHTHS

2019

Kef�laio 1

Idiotimèc kai DiagwniopoÐhsh

Se autì to Kef�laio sumbolÐzoume K eÐte to s¸ma twn pragmatik¸n arijm¸n R, eÐte to

s¸ma twn migadik¸n arijm¸n C. 'Oloi oi pÐnakec eqoun stoiqeÐa sto R sto C, kai ìloioi dianusmatikoÐ q¸roi orÐzontai p�nw apì touc pragmatikoÔc arijmoÔc p�nw apì touc

migadikoÔc arijmoÔc. SumbolÐzoume M(n, K) to sÔnolo twn n × n tetragwnik¸n pin�kwn

me stoiqeÐa sto s¸ma K.

1

2 Grammik 'Algebra I

Di�lexh 1

Idiotimèc kai idiodianÔsmata enìc tetragwnikoÔ pÐnaka

Sto pr¸to mèroc tou maj matoc MEM112 Eisagwg sth Grammik 'Algebra to basikì

antikeÐmeno pou melet same tan h exÐswsh

Ax = b ,

ìpou o �gnwstoc eÐnai to di�nusma x, en¸ sth dexi� pleur� èqoume èna dedomèno di�nusma

b.

Apì th melèth aut c thc exÐswshc kataskeu�same mÐa ploÔsia jewrÐa, pou perigr�fei,

metaxÔ �llwn, th grammik apeikìnish TA : Kn −→ Km : x 7→ Ax. H TA apeikonÐzei ta

stoiqeÐa tou mhdenìqwrou tou pÐnaka A, N (A) sto 0, en¸ apeikonÐzei ta stoiqeÐa tou q¸rou

gramm¸n R(AT ) amfimonos manta sta stoiqeÐa tou q¸rou sthl¸n R(A).

Se autì to Kef�laio ja exet�soume, gia ènan tetragwnikì n× n pÐnaka A, thn exÐswsh

Ax = λx ,

ìpou oi �gnwstoi eÐnai o arijmìc λ kai to di�nusma x. Parathr ste oti to di�nusma emfa-

nÐzetai kai stic dÔo pleurèc thc exÐswshc. AnazhtoÔme dianÔsmata sta opoÐa h apeikìnish

TA dra me ton pio aplì trìpo: ta pollaplasi�zei me ènan arijmì λ, qwrÐc na all�zei th

��dieÔjuns �� touc.

Par�deigma 1.1 O pÐnakac A =

[2 0

1 3

]all�zei th dieÔjunsh tou dianÔsmatoc

[1

0

]:

[2 0

1 3

] [1

0

]=

[2

1

],

en¸ to di�nusma

[0

1

]apl¸c to pollaplasi�zei me ton arijmì 3:

[2 0

1 3

] [0

1

]= 3

[0

1

].

Par�deigma 1.2 JewroÔme ton pÐnaka A =

[5 −1

−1 5

]kai ta dianÔsmata x =

[1

1

]

kai y =

[1

−1

]. ParathroÔme oti

Ax =

[5 −1

−1 5

] [1

1

]=

[4

4

]= 4x

Kef�laio 1 Idiotimèc kai DiagwniopoÐhsh 3

kai

Ay =

[5 −1

−1 5

] [1

−1

]=

[6

−6

]= 6y .

E�n gr�youme opoiod pote di�nusma u ∈ R2 wc grammikì sunduasmì twn x kai y, eÔkola

brÐskoume th dr�sh tou A se autì: e�n u = cx+ dy, tìte

Au = cAx+ dAy

= 4cx+ 6dy.

Ta mh mhdenik� dianÔsmata x pou ikanopoioÔn thn exÐswsh Ax = λx gia k�poio arijmì

λ ∈ K eÐnai, kat� k�poio trìpo, eidik� dianÔsmata tou pÐnaka A: aut� p�nw sta opoÐa o

pollaplasiasmìc me ton A dra me ton aploÔstero trìpo. Gi� autì onom�zontai idiodianÔ-

smata tou pÐnaka (sta agglik� eigenvectors, èqei diathrhjeÐ o germanikìc ìroc wc pr¸to

sunjetikì). Ektìc apì to jewrhtikì endiafèron, gia na katano soume kalÔtera th dr�sh

tou pÐnaka, ta idiodianÔsmata parousi�zoun amètrhtec efarmogèc, se polloÔc kl�douc twn

majhmatik¸n kai �llwn episthm¸n. K�poiec apì autèc tic efarmogèc ja doÔme argìtera.

Orismìc 1.1. JewroÔme tetragwnikì n× n pÐnaka A me stoiqeÐa sto s¸ma K.

Oi arijmoÐ λ ∈ K gia touc opoÐouc up�rqoun mh mhdenikèc lÔseic thc exÐswshc

Ax = λx (1.1)

onom�zontai idiotimèc tou pÐnaka A.

Ta m mhdenik� dianÔsmata x ∈ Kn pou ikanopoioÔn thn exÐswsh 1.1 onom�zontai idiodia-

nÔsmata tou pÐnaka A gia thn idiotim λ.

Drasthriìthta 1.1 Elègxte oti x =

1

0

1

eÐnai èna idiodi�nusma tou pÐnaka

3 −2 0

4 2 −4

1 −1 2

. Se poi� idiotim tou pÐnaka antistoiqeÐ to idiodi�nusma x?

P¸c ja broÔme tic lÔseic thc exÐswshc Ax = λx? ParathroÔme oti gia opoiod pote λ,

to di�nusma 0 eÐnai p�nta mÐa lÔsh. Mac endiafèroun oi m mhdenikèc lÔseic.

Gr�foume thn exÐswsh 1.1 sth morf

Ax− λx = 0

kai eis�goume ton tautotikì pÐnaka I,

Ax− λIx = 0


gia na katal xoume sthn exÐswsh

(A− λI)x = 0 .

Blèpoume oti ta x pou anazhtoÔme brÐskontai sto mhdenoq¸ro tou pÐnaka A−λI. Sunep¸c,

to pr¸to b ma eÐnai na prosdiorÐsoume touc arijmoÔc λ gia touc opoÐouc o mhdenoq¸roc tou

pÐnaka A−λI perièqei m mhdenik� dianÔsmata. Autì sumbaÐnei mìnon ìtan o pÐnakac A−λI

eÐnai idiìmorfoc. H orÐzousa tou pÐnaka mac dÐdei to kat�llhlo krit rio: o pÐnakac A− λI

eÐnai idiìmorfoc e�n kai mìnon e�n det(A− λI) = 0.

Prìtash 1.1 Oi idiotimèc tou n× n pÐnaka A eÐnai oi lÔseic thc exÐswshc

det(A− λI) = 0 . (1.2)

H orÐzousa det(A − λI) eÐnai èna polu¸numo bajmoÔ n me metablht λ. Onom�zetai

qarakthristikì polu¸numo tou pÐnaka A, kai sumbolÐzetai χA. Sunep¸c oi idiotimèc

tou pÐnaka A eÐnai oi rÐzec tou qarakthristikoÔ poluwnÔmou tou A.

E�n λi eÐnai mÐa idiotim tou A, tìte A− λiI èqei mh tetrimmèno mhdenoq¸ro.

Prìtash 1.2 Ta idiodianÔsmata tou A gia thn idiotim λi eÐnai ta mh mhdenik� dianÔsmata

tou mhdenoq¸rou tou pÐnaka A− λiI.

'Oloc o mhdenoq¸roc tou A− λiI onom�zetai idioq¸roc tou A gia thn idiotim λi.

TonÐzoume oti to mhdenikì di�nusma den eÐnai idiodi�nusma. O idiìqwroc tou

A gia thn idiotim λi apoteleÐtai apì to mhdenikì di�nusma kai ìla ta idiodianÔsmata tou A

gia thn idiotim λi.

Drasthriìthta 1.2 BreÐte ta idiodianÔsmata tou pÐnaka A =

4 −3 0

2 −1 0

1 −1 2

gia thn idiotim λ = 1, dhlad ta mh mhdenik� dianÔsmata x pou ikanopoioÔn Ax = x.

Par�deigma 1.3 JewroÔme ton 2× 2 pÐnaka A,

A =

[4 −5

2 −3

].

Gia na upologÐsoume tic idiotimèc, jewroÔme thn orÐzousa

det(A− λI) =

∣∣∣∣∣ 4− λ −5

2 −3− λ

∣∣∣∣∣= −(4− λ)(3 + λ) + 10

= λ2 − λ− 2 .

To polu¸numo χA = λ2 − λ − 2 paragontopoieÐtai, χA = (−1 − λ)(2 − λ). Oi rÐzec tou

poluwnÔmou eÐnai −1 kai 2. 'Ara oi idiotimèc tou pÐnaka A eÐnai

λ1 = −1 kai λ2 = 2 .


Ta idiodianÔsmata gia thn idiotim λ1 = −1, eÐnai oi m mhdenikèc lÔseic thc omogenoÔc

exÐswshc

(A− λ1I)x = 0 ,

dhlad thc exÐswshc [5 −5

2 −2

] [x1

x2

]= 0 .

'Ena idiodi�nusma eÐnai to x =

[1

1

], en¸ o idiìqwroc tou A gia thn idiotim λ1 = −1 eÐnai

o mhdenoq¸roc tou pÐnaka A− λ1I, dhlad o upìqwroc

X1 = {t(1, 1) : t ∈ R} .

Gia thn idiotim λ2 = 2 èqoume, an�loga,

(A− λ2I)x = 0

[2 −5

2 −5

] [x1

x2

]= 0 .

'Ena idiodi�nusma eÐnai to x =

[5

2

], kai o idiìqwroc tou A gia thn idiotim λ2 = 2 eÐnai o

upìqwroc

X2 = {t(5, 2) : t ∈ R} .

ParathroÔme oti se autì to par�deigma oi idiìqwroi tou A gia tic dÔo idiotimèc eÐnai upìqwroi

tou R2 di�stashc 1. Ta dÔo idiodianÔsmata pou br kame eÐnai grammik� anex�rthta, kai

apoteloÔn b�sh tou q¸rou R2

Par�deigma 1.4 JewroÔme ton pÐnaka

B =

3 1 1

2 4 2

−1 −1 1

.Oi idiotimèc tou B eÐnai oi rÐzec tou qarakthristikoÔ poluwnÔmou

χB(λ) =

∣∣∣∣∣∣∣3− λ 1 1

2 4− λ 2

−1 −1 1− λ

∣∣∣∣∣∣∣= (3− λ)(4− λ)(1− λ) + (4− λ)

= (2− λ)2(4− λ) .

'Ara oi idiotimèc eÐnai λ1 = 2 kai λ2 = 4.


Ta idiodianÔsmata gia thn idiotim λ1 = 2 eÐnai oi mh mhdenikèc lÔseic thc exÐswshc 1 1 1

2 2 2

−1 −1 −1

x1

x2

x3

= 0 .

Fèrnoume ton pÐnaka se klimakwt morf , kai èqoume thn exÐswsh 1 1 1

0 0 0

0 0 0

x1

x2

x3

= 0 .

H exÐswsh èqei 2 eleÔjerec metablhtèc. MÐa b�sh tou mhdenìqwrou eÐnai ta dianÔsmata

(−1, 0, 1) kai (−1, 1, 0). O idiìqwroc tou B gia thn idiotim λ1 = 2 eÐnai o

X1 = {s(−1, 0, 1) + t(−1, 1, 0) : s, t ∈ R} .

Ta idiodianÔsmata gia thn idiotim λ2 = 4 eÐnai oi mh mhdenikèc lÔseic thc exÐswshc −1 1 1

2 0 2

−1 −1 −3

x1

x2

x3

= 0 .

Fèrnoume ton pÐnaka se klimakwt morf , kai èqoume thn exÐswsh −1 1 1

0 2 4

0 0 0

x1

x2

x3

= 0 .

'Ena idiodi�nusma tou B gia thn idiotim λ2 = 4 eÐnai to (−1, −2, 1). O idiìqwroc tou B

gia thn idiotim λ2 = 4 eÐnai

X2 = {t(−1, −2, 1) : t ∈ R} .

ParathroÔme oti se autì to par�deigma oi idiìqwroi tou B gia tic dÔo idiotimèc λ1 = 2

kai λ2 = 4eÐnai upìqwroi tou R3 di�stashc 2 kai 1 antÐstoiqa. Br kame trÐa grammik�

anex�rthta idiodianusmata, pou apoteloÔn b�sh tou q¸rou R3.

Drasthriìthta 1.3 O mhdenìqwroc tou pÐnaka A =

[3 4

3 4

]par�getai apì

to di�nusma

[4

−3

]. BreÐte mÐa idiotim kai èna idiodi�nusma gia ton pÐnaka B =[

2 4

3 3

].



C =

1 −2 2

−2 1 2

−2 0 3

.Oi idiotimèc tou C eÐnai oi rÐzec tou poluwnÔmou

χC(λ) =

∣∣∣∣∣∣∣1− λ −2 2

−2 1− λ 2

−2 0 3− λ

∣∣∣∣∣∣∣= (1− λ)2(3− λ) .

'Ara oi idiotimèc eÐnai λ1 = 1 kai λ2 = 3.

Ta idiodianÔsmata gia thn idiotim λ1 = 1 eÐnai oi mh mhdenikèc lÔseic thc exÐswshc 0 −2 2

−2 0 2

−2 0 2

x1

x2

x3

= 0 .

Fèrnoume ton pÐnaka se klimakwt morf , kai èqoume thn exÐswsh −2 0 2

0 −2 2

0 0 0

x1

x2

x3

= 0 .

MÐa lÔsh thc exÐswshc eÐnai h x = (1, 1, 1). 'Ara èna idiodi�nusma tou C gia thn idiotim

λ1 = 1 eÐnai to x = (1, 1, 1). O idiìqwroc tou C gia thn idiotim λ1 = 1 eÐnai o q¸roc

lÔsewn thc exÐswshc,

X1 = {t(1, 1, 1) : t ∈ R} .

Ta idiodianÔsmata gia thn idiotim λ2 = 3 eÐnai lÔseic thc exÐswshc −2 −2 2

−2 −2 2

−2 0 0

x1

x2

x3

= 0 .

'Ena idiodi�nusma tou C gia thn idiotim λ2 = 3 eÐnai to x = (0, 1, 1). O idiìqwroc tou C

gia thn idiotim λ2 = 3 eÐnai

X2 = {t(0, 1, 1) | t ∈ R} .

ParathroÔme oti se autì to par�deigma, stic dÔo diaforetikèc idiotimèc antistoiqoÔn

mìno dÔo grammik� anex�rthta idiodianÔsmata, ta opoÐa den par�goun ìlo to q¸ro R3.


UpenjumÐzoume to Jemeli¸dec Je¸rhma thc 'Algebrac, (Shmei¸seic EpÐpedo kai Q¸roc,

Kef�laio 3), sÔmfwna me to opoÐo gia èna polu¸numo me pragmatikoÔc migadikoÔc sunte-

lestèc, bajmoÔ n,

p(z) = anzn + · · ·+ a1z + a0 , an 6= 0, a0, . . . , an ∈ C ,

up�rqoun n migadikoÐ arijmoÐ w1, . . . , wn, (oi rÐzec tou poluwnÔmou p(z)), ìqi upoqrewtik�

diaforetikoÐ, tètoioi ¸ste

p(z) = an(z − w1)(z − w2) · · · (z − wn) .

E�n akribwc k apì touc migadikoÔc arijmoÔc w1, . . . , wn eÐnai Ðsoi me w, lème oti w eÐnai rÐza

tou p(z) me pollaplìthta k. Tìte (z −w)k diaireÐ to polu¸numo p(z), all� (z −w)k+1

den to diaireÐ.

Sto Par�deigma 1.4, to qarakthristikì polu¸numo χB(λ) = (2 − λ)2(4 − λ) èqei rÐzec

λ1 = 2 me pollaplìthta 2, kai λ2 = 4 me pollaplìthta 1. Lème oti h idiotim λ1 = 2

èqei algebrik pollaplìthta 2, en¸ h idiotim λ2 = 4 èqei algebrik pollaplìthta 1. Se

aut n thn perÐptwsh h algebrik pollaplìthta k�je idiotim c eÐnai Ðsh me th di�stash tou

antÐstoiqou idiìqwrou.

Antijètwc, sto Par�deigma 1.5, to qarakthristikì polu¸numo χC(λ) = (1− λ)2(3− λ)

h idiotim λ1 = 1 èqei algebrik pollaplìthta 2, all� o antÐstoiqoc idiìqwroc tou pÐnaka

C èqei di�stash 1.

Genikìtera, orÐzoume thn algebrik pollaplìthta mÐac idiotim c na eÐnai h polla-

plìthta thc idiotim c wc rÐzac tou qarakthristikoÔ poluwnÔmou, en¸ orÐzoume th gew-

metrik pollaplìthta thc idiotim c na eÐnai h di�stash tou idiìqwrou pou antistoiqeÐ

sthn idiotim . Ja doÔme oti aut h di�krish metaxÔ thc algebrik c kai thc gewmetrik c

pollaplìthtac mÐac idiotim c èqei meg�lh shmasÐa.


Di�lexh 2Oi rÐzec enìc poluwnÔmou me pragmatikoÔc suntelestèc mporeÐ na eÐnai migadikoÐ arijmoÐ.

'Etsi, ènac pÐnakac me pragmatik� stoiqeÐa mporeÐ na èqei migadikèc idiotimèc, ìpwc ja doÔme

sto epìmeno par�deigma.


A =

4 0 3

0 2 0

−3 0 4

.Oi idiotimèc tou A eÐnai oi rÐzec tou

χA(λ) =

∣∣∣∣∣∣∣4− λ 0 3

0 2− λ 0

−3 0 4− λ

∣∣∣∣∣∣∣= (λ2 − 8λ+ 25)(2− λ) .

To polu¸numo èqei mÐa pragmatik rÐza, λ1 = 2. Oi �llec dÔo rÐzec eÐnai migadikèc,

λ2 = 4 + 3i kai λ3 = 4− 3i .

Gia thn idiotim λ1 = 2 èqoume thn exÐswsh 2 0 3

0 0 0

−3 0 2

x1

x2

x3

= 0

apì thn opoÐa brÐskoume oti èna idiodi�nusma tou A gia thn idiotim λ1 = 2 eÐnai to x =

(0, 1, 0).

Gia thn idiotim λ2 = 4 + 3i èqoume thn exÐswsh −3i 0 3

0 −2− 3i 0

−3 0 −3i

x1

x2

x3

= 0

Fèrnoume ton pÐnaka se klimakwt morf kai èqoume thn exÐswsh −3i 0 3

0 −2− 3i 0

0 0 0

x1

x2

x3

= 0

apì thn opoÐa brÐskoume oti èna idiodi�nusma tou A gia thn idiotim λ2 = 4 + 3i eÐnai to

x = (−i, 0, 1).


Gia thn idiotim λ3 = 4− 3i èqoume thn exÐswsh 3i 0 3

0 −2 + 3i 0

−3 0 3i

x1

x2

x3

= 0

apì thn opoÐa brÐskoume oti èna idiodi�nusma tou A gia thn idiotim λ3 = 4 − 3i eÐnai to

x = (i, 0, 1).

E�n jewr soume ton pÐnaka A p�nw apì touc pragmatikoÔc arijmoÔc, autìc èqei mìno

mÐa idiotim , λ1 = 2, kai o antÐstoiqoc idiìqwroc eÐnai o

X1 = {t(0, 1, 0) : t ∈ R} .

E�n jewr soume ton pÐnaka A p�nw apì touc migadikoÔc arijmoÔc tìte o idiìqwroc gia

thn idiotim λ1 = 2 eÐnai o

X1 = {t(0, 1, 0) : t ∈ C} ,

o idiìqwroc gia thn idiotim λ2 = 4 + 3i eÐnai o

X2 = {t(−i, 0, 1) : t ∈ C}

kai o idiìqwroc gia thn idiotim λ3 = 4− 3i eÐnai o

X3 = {t(i, 0, 1) : t ∈ C} .

Anakefalai¸noume th diadikasÐa gia ton upologismì twn idiotim¸n kai twn idiodianusm�-

twn enìc n× n pÐnaka

1. UpologÐzoume thn orÐzousa tou pÐnaka A − λI. Aut eÐnai èna polu¸numo bajmoÔ n

wc proc th metablht λ, to qarakthristikì polu¸numo tou A.

2. BrÐskoume tic rÐzec tou qarakthristikoÔ polu¸numou. Autèc eÐnai oi idiotimèc tou A.

3. Gia k�je idiotim λi, brÐskoume tic lÔseic thc omogenoÔc exÐswshc

(A− λiI)x = 0 .

K�je mh mhdenik lÔsh eÐnai èna idiodi�nusma tou pÐnaka A gia thn idiotim λi, en¸ to

sÔnolo ìlwn twn lÔsewn eÐnai o idiìqwroc tou A gia thn idiotim λi.

Drasthriìthta 1.4 UpologÐste to qarakthristikì polu¸numo kai breÐte tic

idiotimèc kai ta idiodianÔsmata tou pÐnaka1

A =

[7 −15

2 −4

].


En antijèsei me thn perÐptwsh thc lÔshc tou sust matoc Ax = b me apaloif Gauss,

h diadikasÐa pou perigr�foume ed¸ den dÐdei ènan algìrijmo gia ton analutikì upologismì

twn idiotim¸n kai twn idiodianusm�twn. To prìblhma brÐsketai sto b ma 2. En¸ gnwrÐzoume

oti k�je polu¸numo bajmoÔ n èqei n rÐzec (sto sÔnolo C twn migadik¸n arijm¸n), gia

polu¸numa bajmoÔ n ≥ 5 den eÐnai dunatìn na brejeÐ analutikìc tÔpoc gia ton upologismì

touc (ìpwc o tÔpoc twn riz¸n thc deuterob�jmiac exÐswshc)2.

Par� ìlo pou den up�rqei analutikìc tÔpoc pou na dÐdei tic rÐzec sth genik perÐptwsh,

se pollèc eidikèc peript¸seic mporoÔme na tic prosdiorÐsoume analutik�, mporoÔme na tic

proseggÐsoume arijmhtik�. Ja doÔme oti mporoÔme na èqoume k�poia plhroforÐa gia tic

idiotimèc akìma kai qwrÐc na tic upologÐsoume.

To �jroisma twn diag¸niwn stoiqeÐwn enìc pÐnaka onom�zetai Ðqnoc tou pÐnaka (trace)

kai sumbolÐzetai trA.

Prìtash 1.3 JewroÔme ènan n × n pÐnaka A p�nw apì touc migadikoÔc arijmoÔc. Me-

tr¸ntac thn algebrik pollaplìthta, o pÐnakac èqei n idiotimèc, λ1, λ2, . . . , λn, ìqi upo-

qrewtik� ìlec diaforetikèc.

1. To �jroisma twn idiotim¸n tou A eÐnai Ðso me to Ðqnoc tou pÐnaka,

λ1 + λ2 + · · ·+ λn = trA

2. To ginìmeno twn idiotim¸n tou A eÐnai Ðso me thn orÐzousa tou pÐnaka,

λ1 λ2 · · ·λn = detA .

Apìdeixh.

1. E�n λ1, . . . , λn eÐnai oi idiotimèc tou pÐnaka A, èqoume

det(A− λI) = (λ1 − λ) · · · (λn − λ) .

SugkrÐnoume touc ìrouc t�xewc n− 1 sta dÔo polu¸numa. Oi ìroi stouc opoÐouc to

λ emfanÐzetai sth dÔnamh n− 1 sthn orÐzousa∣∣∣∣∣∣∣∣∣∣a11 − λ a12 · · · a1n

a21 a22 − λ · · · a2n...

.... . .

...

an1 an2 · · · ann − λ

∣∣∣∣∣∣∣∣∣∣prèpei na proèrqontai apì ìrouc thc orÐzousac pou eÐnai ginìmeno toul�qiston n− 1

stoiqeÐwn sth diag¸nio tou pÐnaka. All� ènac ìroc thc orÐzousac den mporeÐ na

perièqei perissìtera apì èna stoiqeÐo apì k�je st lh kai apì k�je gramm tou pÐnaka.

1Ta dianÔsmata pou ja breÐte prèpei na eÐnai mh mhdenik� pollapl�sia twn (5, 2) kai (3, 1).2Autì eÐnai to perieqìmeno thc jewrÐac Galois (thn opoÐa mporeÐte na melet sete sto m�jhma JewrÐa

Swm�twn), mÐac polÔ endiafèrousac jewrÐac pou dhmioÔrghse ènac akìmh pio endiafèrwn �njrwpoc.


Sunep¸c, o monadikìc ìroc pou perièqei to ginìmeno n− 1 diag¸niwn stoiqeÐwn, eÐnai

to ginìmeno ìlwn twn diag¸niwn stoiqeÐwn,

(a11 − λ)(a22 − λ) · · · (ann − λ) .

O ìroc t�xewc n− 1 autoÔ tou poluwnÔmou eÐnai

a11λn−1 + a22λ

n−1 + · · ·+ annλn−1 = (a11 + · · ·+ ann)λn−1 .

Apì thn �llh pleur� o ìroc t�xewc n−1 tou poluwnÔmou (λ1−λ)(λ2−λ) · · · (λn−λ)

eÐnai

(λ1 + λ2 + · · ·+ λn)λn−1 .

SumperaÐnoume oti

λ1 + λ2 + · · ·+ λn = a11 + a22 + · · ·+ ann = trA .

2. Exet�zoume tou stajeroÔc ìrouc twn poluwnÔmwn

det(A− λI) = (λ1 − λ) · · · (λn − λ) .

Sth dexi� pleur�, o stajerìc ìroc eÐnai λ1λ2 · · ·λn. Sthn arister pleur� o stajerìcìroc eÐnai h tim tou poluwnÔmou gia λ = 0, dhlad detA. 'Ara

λ1λ2 · · ·λn = detA .

�

Drasthriìthta 1.5 UpologÐste tic idiotimèc tou pÐnaka A =

[2 1

9 2

], dhlad

tic rÐzec tou poluwnÔmou det(A− λI) =

∣∣∣∣∣ 2− λ 1

9 2− λ

∣∣∣∣∣.EpalhjeÔste oti to �jroisma twn idiotim¸n eÐnai Ðso me to Ðqnoc tou pÐnaka kai to

ginìmeno twn idiotim¸n eÐnai Ðso me thn orÐzousa tou pÐnaka.

Drasthriìthta 1.6 UpologÐste tic migadikèc idiotimèc tou pÐnaka B =[1 1

−2 3

].

EpalhjeÔste oti to �jroisma twn idiotim¸n eÐnai Ðso me to Ðqnoc tou pÐnaka kai to


Gia k�je idiotim breÐte èna idiodi�nusma tou B sto C2. (Parathr ste oti (1+i)(1−i) = 2.)


DiagwniopoÐhsh

Sto Par�deigma 1.2 parathr same oti e�n gr�youme èna di�nusma wc grammikì sunduasmì

idiodianusm�twn tou A, tìte mporoÔme eÔkola na perigr�youme th dr�sh tou A se autì to

di�nusma. AfoÔ

[2

4

]= 3

[1

1

]− 1

[1

−1

],

[5 −1

−1 5

][2

4

]= 3

[5 −1

−1 5

][1

1

]− 1

[5 −1

−1 5

][1

−1

]

= 12

[1

1

]− 6

[1

−1

].

E�n ènac n × n pÐnakac èqei n grammik� anex�rthta idiodianÔsmata, tìte k�je di�nusma

tou Kn gr�fetai wc grammikìc sunduasmìc twn idiodianusm�twn, kai o pollaplasiasmìc o-

poioud pote dianÔsmatoc me ton pÐnaka ekfr�zetai me autì ton trìpo. To akìloujo je¸rhma

dÐdei mia pio akrib diatÔpwsh aut c thc idèac.

Je¸rhma 1.4 Upojètoume oti o n × n pÐnakac A èqei n grammik� anex�rthta idiodia-

nÔsmata x1, . . . , xn. JewroÔme to pÐnaka R, o opoÐoc èqei wc st lec ta idiodianÔsmata

x1, . . . , xn. Tìte o pÐnakac

Λ = R−1AR

eÐnai diag¸nioc, kai ta stoiqeÐa sth diag¸nio eÐnai oi idiotimèc λ1, . . . , λn tou A.

Dhlad

Λ = R−1AR =

λ1 0

. . .

0 λn

kai

A = RΛR−1 =

...

...

x1 · · · xn...

...

λ1 0

. . .

0 λn

......

x1 · · · xn...

...

−1

.

Apìdeixh. H j-st lh tou pÐnaka AR eÐnai to di�nusma Axj = λjxj . 'Ara

AR =

...

...

λ1x1 · · · λnxn...

...

.

H j-st lh tou R−1(AR) eÐnai h j-st lh tou AR pollaplasiasmènh me ton pÐnaka R−1.

All� h j-st lh tou AR eÐnai h j-st lh tou R pollaplasiasmènh epÐ λj . 'Ara h j-st lh tou


R−1(AR) eÐnai λj × (j-st lh tou R−1R), dhlad λjej . SumperaÐnoume oti

R−1AR =

...

...

λ1e1 · · · λnen...

...

,dhlad o diag¸nioc pÐnakac me tic idiotimèc λ1, . . . , λn sth diag¸nio.

�

Par�deigma 1.7 Sta ParadeÐgma 1.4 eÐdame oti o pÐnakac

B =

3 1 1

2 4 2

−1 −1 1

.èqei trÐa grammik� anex�rthta idiodianÔsmata, (−1, 0, 1), (−1, 1, 0) kai (−1, −2, 1). Jew-

roÔme ton pÐnaka R me st lec aut� ta idiodianÔsmata,

R =

−1 −1 −1

0 1 −2

1 0 1

.'Eqoume

BR =

3 1 1

2 4 2

−1 −1 1

−1 −1 −1

0 1 −2

1 0 1

=

−2 −2 −4

0 2 −8

2 0 4

,kai

R

2 0 0

0 2 0

0 0 4

=

−2 −2 −4

0 2 −8

2 0 4

.Prosèxte oti h di�taxh twn idiodianusm�twn ston pÐnaka R eÐnai h Ðdia me th di�taxh twn

antÐstoiqwn idiotim¸n ston pÐnaka Λ.

Drasthriìthta 1.7 Sth Drasthriìthta 1.4 br kate tic idiotimèc kai ta idio-

dianÔsmata tou pÐnaka A =

[7 −15

2 −4

]. B�lte ta idiodianÔsmata pou br kate wc

st lec tou pÐnaka R kai upologÐste touc pÐnakec R−1 kai R−1AR.

Drasthriìthta 1.8 Jewr ste ton pÐnaka R pou èqei wc st lec ta idiodianÔ-

smata tou pÐnaka B pou upologÐsate sth Drasthriìthta 1.6, kai ton diag¸nio pÐnaka

Λ pou èqei tic idiotimèc sthn antÐstoiqh jèsh sth diag¸nio.

EpalhjeÔste oti BR = RΛ.

'Enac pÐnakac A gia ton opoÐo up�rqei antistrèyimoc pÐnakac R tètoioc ¸ste o R−1AR na

eÐnai diag¸nioc onom�zetai diagwniopoi simoc. Ja deÐxoume oti idiodianÔsmata pou anti-

stoiqoÔn se diaforetikèc idiotimèc eÐnai grammik� anex�rthta. Autì sunep�getai oti e�n ènac


n×n pÐnakac èqei n diaforetikèc idiotimèc, tìte èqei n grammik� anex�rthta idiodianÔsmata,

kai sunep¸c eÐnai diagwniopoi simoc.

L mma 1.5 Ta idiodianÔsmata pou antistoiqoÔn se diaforetikèc idiotimèc eÐnai grammik�

anex�rthta.

Apìdeixh. E�n λ1, . . . , λm eÐnai oi diaforetikèc idiotimèc tou A, kai ta antÐstoiqa idio-

dianÔsmata v1, . . . , vm eÐnai grammik� exarthmèna, tìte up�rqei k, me 1 < k ≤ m, tètoio

¸ste {v1, . . . , vk−1} eÐnai grammik� anex�rthta, all� v1, . . . , vk eÐnai grammik� exarthmèna

kai mporoÔme na gr�youme

vk = a1v1 + · · ·+ ak−1vk−1 . (1.3)

Pollaplasi�zontac tic dÔo pleurèc thc 1.3 me A èqoume

Avk = a1Av1 + · · ·+ ak−1Avk−1

kai afoÔ k�je vi eÐnai idiodi�nusma gia thn idiotim λi,

λkvk = a1λ1v1 + · · ·+ ak−1λk−1vk−1 (1.4)

Pollaplasi�zoume thn 1.3 me λk, kai thn afairoÔme apì thn 1.4:

0 = a1(λ1 − λk)v1 + · · ·+ ak−1(λk−1 − λk)vk−1 .

Ef� ìson ta v1, . . . , vk−1 eÐnai grammik� anex�rthta, ai(λi−λk) = 0 gia k�je i = 1, . . . , k−1,

all� λi−λk 6= 0, kai sunep¸c a1 = · · · = ak−1 = 0. All� tìte, apì thn 1.3, vk = 0, �topo.

SumperaÐnoume oti to sÔnolo {v1, . . . , vm} eÐnai grammik� anex�rthto.

�


Ask seic 1

'Askhsh 1.1 BreÐte tic idiotimèc kai ta idiodianÔsmata tou pÐnaka

A =

[1 −1

2 4

].

EpalhjeÔste oti to �jroisma twn idiotim¸n eÐnai Ðso me to Ðqnoc tou pÐnaka, kai to


'Askhsh 1.2 E�n B = A − 7I, ìpou A eÐnai o pÐnakac thc 'Askhshc 1.1, breÐte

tic idiotimèc kai ta idiodianÔsmata tou B. Pwc sqetÐzontai me aut� tou A?

'Askhsh 1.3 D¸ste èna par�deigma gia na deÐxete oti oi idiotimèc all�zoun ìtan

afairèsoume pollapl�sio mÐac gramm c apì mÐa �llh. Exhg ste giatÐ e�n to 0 eÐnai

mÐa apì tic idiotimèc, aut den all�zei.

'Askhsh 1.4 BreÐte tic idiotimèc kai ta idiodianÔsmata twn pin�kwn

A =

3 4 2

0 1 2

0 0 0

kai B =

0 0 2

0 2 0

2 0 0

.Elègxte oti to �jroisma twn idiotim¸n eÐnai Ðso me to Ðqnoc, kai to ginìmeno me thn

orÐzousa.

'Askhsh 1.5 Upojètoume oti λ eÐnai idiotim tou antistrèyimou pÐnaka A, kai x

eÐnai idiodi�nusma: Ax = λx. DeÐxte oti x eÐnai epÐshc idiodi�nusma tou antistrìfou

A−1, kai breÐte thn antÐstoiqh idiotim .

'Askhsh 1.6 DeÐxte oti oi idiotimèc tou an�strofou pÐnaka AT eÐnai Ðsec me tic

idiotimèc tou A.

'Askhsh 1.7 Kataskeu�ste 2×2 pÐnakec A kai B, tètoiouc ¸ste oi idiotimèc tou

AB den eÐnai Ðsec me ta ginìmena twn idiotim¸n tou A kai tou B, kai oi idiotimèc tou

A+B den eÐnai Ðsec me ta ajroÐsmata twn idiotim¸n.


'Askhsh 1.8 Upojètoume oti o 3×3 pÐnakac A èqei idiotimèc 0, 3, 5 me antÐstoiqa

idiodianÔsmata u, v, w.

1. BreÐte mÐa b�sh tou mhdenoq¸rou tou A, kai mÐa b�sh tou q¸rou sthl¸n tou

A.

2. BreÐte mÐa lÔsh thc exÐswshc Ax = v+w. BreÐte ìlec tic lÔseic thc exÐswshc.

3. DeÐxte oti h exÐswsh Ax = u den èqei lÔseic.

'Askhsh 1.9 BreÐte tic idiotimèc kai ta idiodianÔsmata twn pin�kwn

A =

[3 4

4 −3

]kai B =

[a b

b a

].

'Askhsh 1.10 K�je pÐnakac met�jeshc af nei to di�nusma x = (1, 1, . . . , 1)

amet�blhto. 'Ara èqei mÐa idiotim λ = 1. BreÐte �llec dÔo idiotimèc gia touc pÐnakec

P =

0 1 0

0 0 1

1 0 0

kai R =

0 0 1

0 1 0

1 0 0

.'Askhsh 1.11 Diagwniopoi ste touc akìloujouc pÐnakec (dhlad breÐte R tè-

toiouc ¸ste R−1AR eÐnai diag¸nioc pÐnakac):[1 1

1 1

]kai

[2 1

0 0

].

'Askhsh 1.12 BreÐte ìlec tic idiotimèc kai idiodianÔsmata tou pÐnaka

A =

1 1 1

1 1 1

1 1 1

kai gr�yte dÔo diaforetikoÔc pÐnakec R pou diagwniopoioÔn ton A.

'Askhsh 1.13 PoioÐ apì touc akìloujouc pÐnakec den mporoÔn na diagwniopoih-

joÔn?

A1 =

[2 −2

2 −2

]A2 =

[2 0

2 −2

]A3 =

[2 0

2 2

].


Di�lexh 3

Efarmogèc thc diagwniopoÐhshc

H diagwniopoÐhsh eÐnai mÐa teqnik pou qrhsimopoieÐtai se polloÔc kl�douc twn majhma-

tik¸n, all� kai se �llec epist mec. Stic epìmenec paragr�fouc ja doÔme k�poia apl�

paradeÐgmata.

Upologismìc dun�mewn enìc pÐnaka

E�n A, B kai R eÐnai n × n tetragwnikoÐ pÐnakec, R eÐnai antistrèyimoc kai A = RBR−1,

parathroÔme oti

A2 = RBR−1RBR−1 = RB2R−1

A3 = A2A = RB2R−1RBR−1 = RB3R−1

kai mporoÔme na apodeÐxoume me epagwg oti gia k�je k,

Ak = RBkR−1 .

Drasthriìthta 1.9 ApodeÐxte me epagwg sto k oti e�n A = RBR−1, tìte

gia k�je k

Ak = RBkR−1 .

Katagr�yte analutik� ìlh th diadikasÐa thc majhmatik c epagwg c, sÔmfwna me to

upìdeigma sto EpÐpedo kai Q¸roc, Kef�laio 3.

E�n o 2× 2 pÐnakac D eÐnai diag¸nioc, D =

[d1 0

0 d2

], tìte

D2 =

[d1 0

0 d2

][d1 0

0 d2

]=

[d21 0

0 d22

]

D3 =

[d21 0

0 d22

][d1 0

0 d2

]=

[d31 0

0 d32

].

Gia k�je n× n diag¸nio pÐnaka D mporoÔme na apodeÐxoume me epagwg oti gia k�je k,d1 0

. . .

0 dn

k

=

dk1 0

. . .

0 dkn

.Drasthriìthta 1.10 ApodeÐxte me epagwg sto k oti

d1 0. . .

0 dn

k

=

dk1 0

. . .

0 dkn

.


E�n o n × n tetragwnikìc pÐnakac A èqei n grammik� anex�rthta idiodianÔsmata, tìte

sÔmfwna me to Je¸rhma 1.4, e�n R eÐnai o antistrèyimoc pÐnakac me ta idiodianÔsmata wc

st lec, tìte R−1AR eÐnai diag¸nioc pÐnakac me tic idiotimèc sth diag¸nio. Sunep¸c, gia

k�je fusikì arijmì k,

Ak = R

λk1 0

. . .

0 λkn

R−1 .Par�deigma 1.8 Sta ParadeÐgmata 1.4 kai 1.7 eÐdame oti o pÐnakac

B =

3 1 1

2 4 2

−1 −1 1

.èqei trÐa grammik� anex�rthta dianÔsmata (−1, 0, 1), (−1, 1, 0) kai (−1, −2, 1), kai oti

BR =

3 1 1

2 4 2

−1 −1 1

−1 −1 −1

0 1 −2

1 0 1

=

−1 −1 −1

0 1 −2

1 0 1

2 0 0

0 2 0

0 0 4

= RΛ .

UpologÐzoume ton antÐstrofo, R−1,

R−1 =

−1 −1 −1

0 1 −2

1 0 1

−1

=1

2

1 1 3

−2 0 −2

−1 −1 −1

.T¸ra gia na broÔme opoiad pote dÔnamh tou B arkeÐ na upologÐsoume to ginìmeno

Bk = RΛkR−1

=1

2

−1 −1 −1

0 1 −2

1 0 1

2k 0 0

0 2k 0

0 0 4k

1 1 3

−2 0 −2

−1 −1 −1

=1

2

−2k −2k −4k

0 2k −2(4k)

2k 0 4k

1 1 3

−2 0 −2

−1 −1 −1

=1

2

2k + 4k −2k + 4k −2k + 4k

2(−2k + 4k) 2(4k) 2(−2k + 4k)

2k − 4k 2k − 4k 3(2k)− 4k

.Drasthriìthta 1.11 Efarmìste th diadikasÐa Gauss – Jordan ston epekteta-

mèno pÐnaka −1 −1 −1

... 1 0 0

0 1 −2... 0 1 0

1 0 1... 0 0 1

gia na upologÐsete ton antÐstrofo pÐnaka R−1.


Drasthriìthta 1.12 Gia ton pÐnaka A =

[7 −15

2 −4

]thc Drasthriìthtac 1.7,

upologÐste ton pÐnaka A10.

UpologÐste thn analutik èkfrash3 gia ton pÐnaka Ak.

Exis¸seic Diafor¸n

MÐa exÐswsh diafor¸n eÐnai mÐa exÐswsh pou sundèei touc ìrouc mÐac akoloujÐac me

prohgoÔmenouc ìrouc thc akoloujÐac. Gia par�deigma, jewr ste thn akoloujÐa xk pou

orÐzetai apì th sqèsh

xk+1 = 5xk − 1 , me arqik sunj kh x0 = 1 .

Apì autèc tic sqèseic brÐskoume

x1 = 5 · 1− 1 = 4

x2 = 5 · 4− 1 = 19

x3 = 5 · 19− 1 = 94

To zhtoÔmeno eÐnai na broÔme mÐa èkfrash gia to genikì ìro xk thc akoloujÐac.

MÐa idiaÐtera apl perÐptwsh exÐswshc diafor¸n eÐnai h

xk+1 = axk , gia a ∈ R , me arqik sunj kh x0 = c .

Se aut thn perÐptwsh eÔkola deÐqnoume me epagwg oti xk eÐnai mÐa gewmetrik akoloujÐa,

xk = akc.

Par�deigma 1.9 H akoloujÐa Fibonacci orÐzetai apì thn exÐswsh diafor¸n

Fk+1 = Fk + Fk−1 , me arqikèc sunj kec F1 = 1 , F0 = 0 . (1.5)

Oi pr¸toi ìroi thc akoloujÐac Fibonacci eÐnai 0, 1, 1, 2, 3, 5, 8, 13, 21, . . .. Gia na upologÐ-

soume ton genikì ìro thc akoloujÐac jewroÔme thn akoloujÐa dianusm�twn uk = (Fk, Fk−1).

Tìte h exÐswsh diafor¸n 1.5 mporeÐ na ekfrasteÐ wc mÐa dianusmatik exÐswsh diafor¸n,

uk+1 =

[Fk+1

Fk

]=

[1 1

1 0

][Fk

Fk−1

]= Auk−1

gia ton pÐnaka A =

[1 1

1 0

]me arqikèc sunj kec u1 =

[1

0

].

Genikìtera, jewroÔme n akoloujÐec x1 = (x1, k), . . . , xn = (xn, k) kai to di�nusma uk =

(x1, k, . . . , xn, k) ∈ Rn. 'Ena grammikì sÔsthma exis¸sewn diafor¸n eÐnai mÐa sqèsh

uk+1 = Auk , me arqikèc sunj kec u0 = (x1, 0, . . . , xn, 0)

3Prèpei na breÐte

[−5 + 6(2k) 15− 15(2k)

−2 + 2(2k) 6− 5(2k)

].


ìpou A eÐnai ènac n× n pÐnakac.

Prìtash 1.6 H lÔsh tou sust matoc exis¸sewn diafor¸n

uk+1 = Auk , me arqikèc sunj kec u0 = (x1, 0, . . . , xn, 0) (1.6)

eÐnai

uk = Aku0 .

Apìdeixh. Gia k = 1, apì thn 1.6, èqoume u1 = Au0. Upojètoume oti uk = Aku0. Tìte,

apì thn 1.6 èqoume uk+1 = Auk = A(Aku0) = Ak+1u0. Apì to AxÐwma thc Epagwg c,

sumperaÐnoume oti gia k�je fusikì arijmì k, uk = Aku0.

�

Par�deigma 1.10 Oi ìroi thc akoloujÐac Fibonacci dÐdontai apì thn èkfrash uk =

Aku0, dhlad [Fk+1

Fk

]=

[1 1

1 0

]k [1

0

].

Gia na upologÐsoume tic dun�meic Ak, exet�zoume e�n o pÐnakac eÐnai diagwniopoi simoc.

BrÐskoume oti o pÐnakac A èqei qarakthristikì polu¸numo λ2 − λ − 1, to opoÐo èqei rÐzec

tic dÔo idiotimèc, λ1 =1 +√

5

2kai λ2 =

1−√

5

2. Ta idiodianÔsmata tou pÐnaka A gia

tic idiotimèc λi, i = 1, 2 eÐnai oi lÔseic thc exÐswshc

[1− λi 1

1 −λi

][y1

y2

]= 0, dhlad [

λi

1

].

Ekfr�zoume thn arqik sunj kh u0 =

[1

0

]wc grammikì sunduasmì twn idiodianusm�-

twn. Oi suntelestèc c1 kai c2 eÐnai oi lÔseic thc exÐswshc[λ1 λ2

1 1

][c1

c2

]=

[1

0

].

BrÐskoume c1 = 1√5kai c2 = − 1√

5.

'Ara [Fk+1

Fk

]=

[1 1

1 0

]k [1

0

]

=

[1 1

1 0

]k(1√5

[λ1

1

]− 1√

5

[λ2

1

])

=1√5

[ 1 1

1 0

]k [λ1

1

]−

[1 1

1 0

]k [λ2

1

]


=1√5

(λk1

[λ1

1

]− λk2

[λ2

1

]).

Katal goume sthn analutik èkfrash gia ton genikì ìro thc akoloujÐac Fibonacci,

Fk =1√5

(λk1 − λk2) =1√5

(1 +√

5

2

)k−

(1−√

5

2

)k .

Parathr ste oti oi ìroi thc akoloujÐac Fibonacci eÐnai fusikoÐ arijmoÐ. 'Ara sto telikì

apotèlesma ja apaleifjoÔn oi tetragwnikèc rÐzec.

Ac doÔme èna par�deigma me pio aploÔc arijmoÔc, gia na epikentrwjoÔme sth diadikasÐa.

Par�deigma 1.11 JewroÔme to sÔsthma exis¸sewn diafor¸n

xk+1 = 7xk − 15yk

yk+1 = 2xk − 4yk ,

me arqikèc sunj kec x0 = 1, y0 = 1. Ekfr�zoume to sÔsthma wc mÐa dianusmatik exÐswsh

uk+1 = Auk, me uk =

[xk

yk

]kai A =

[7 −15

2 −4

].

Sth Drasthriìthta 1.4 br kame oti o pÐnakac A =

[7 −15

2 −4

]èqei qarakthristikì

polu¸numo χA(λ) = λ2 − 3λ+ 2 kai idiotimèc λ1 = 1, λ2 = 2, me antÐstoiqa idiodianÔsmata[5

2

]kai

[3

1

]. 'Ara

[7 −15

2 −4

]k [5

2

]=

[5

2

]kai

[7 −15

2 −4

]k [3

1

]=

[3(2k)

1(2k)

].

E�n jèsoume ta idiodianÔsmata wc st lec tou pÐnaka R, kai tic antÐstoiqec idiotimèc sth

diag¸nio tou pÐnaka Λ, èqoume Ak = RΛkR−1, dhlad ton pÐnaka pou upologÐsame sth

Drasthriìthta 1.12. Pollaplasi�zoume to dÐanusma twn arqik¸n sunjhk¸n me autìn ton

pÐnaka kai brÐskoume th lÔsh tou sust matoc exis¸sewn diafor¸n,[xk

yk

]=

[7 −15

2 −4

]k [1

1

]=

[10− 9(2k)

4− 3(2k)

].

'Enac �lloc trìpoc na katal xoume sto Ðdio apotèlesma eÐnai na ekfr�soume to di�nusma

twn arqik¸n sunjhk¸n u0 =

[1

1

]wc grammikì sunduasmì twn idiodianusm�twn, lÔnontac

to sÔsthma [1

1

]=

[5 3

2 1

][c

d

].


BrÐskoume c = 2, d = −3. AntikajistoÔme tic arqikèc sunj kec

[1

1

]= 2

[5

2

]−3

[3

1

]sth sqèsh uk = Aku0 kai èqoume[

xk

yk

]=

[7 −15

2 −4

]k(2

[5

2

]− 3

[3

1

])

= 2(1k)

[5

2

]− 3(2k)

[3

1

]

=

[10− 9(2k)

4− 3(2k)

].


Di�lexh 4

Markobianèc diadikasÐec

MÐa Markobian diadikasÐa ( Markobian alussÐda) apoteleÐtai apì ènan stajerì

plhjusmì, ta �toma tou opoÐou brÐskontai se diaforetikèc katast�seic. Se k�je b ma thc

diadikasÐac ta �toma tou plhjusmoÔ mporeÐ na metakinhjoÔn apì mÐa kat�stash se mÐa �llh.

IdiaÐtero qarakthristikì mÐac Markobian c diadikasÐac eÐnai oti h pijanìthta na metakinhjeÐ

èna �tomo sto epìmeno b ma se mÐa kat�stash, exart�tai mìnon apì thn kat�stash sthn

opoÐa brÐsketai, kai ìqi apì to ti èqei sumbeÐ se prohgoÔmena b mata.

Ta dedomèna mÐac Markobian c diadikasÐac me n diaforetikèc katast�seic katagr�fontai

se ènan n×n pÐnaka met�bashc tou opoÐou to stoiqeÐo sthn i gramm kai sthn j st lh

katafr�fei thn pijanìthta èna �tomo pou brÐsketai sthn kat�stash j, sto epìmeno b ma

na metakinhjeÐ sthn kat�stash i. AfoÔ ta stoiqeÐa tou pÐnaka katagr�foun pijanìthtec,

eÐnai arijmoÐ metaxÔ 0 kai 1. 'Ola ta �toma pou brÐskontai se mÐa kat�stash, sto epìmeno

b ma ja prèpei na brÐskontai p�li se mÐa kat�stash, thn Ðdia diaforetik . Sunep¸c to

�jroisma twn pijanot twn se k�je st lh eÐnai 1.

'Enac Markobianìc pÐnakac eÐnai ènac n × n tetragwnikìc pÐnakac pou den perièqei ar-

nhtikoÔc arijmoÔc kai to �jroisma twn stoiqeÐwn k�je st lhc eÐnai 1. 'Ena di�nusma

katanom c eÐnai èna di�nusma ston Rn, tou opoÐou to stoiqeÐo sth jèsh i katagr�fei thn

pijanìthta èna �tomo tou plhjusmoÔ na brÐsketai sthn kat�stash i.

Par�deigma 1.12 JewroÔme oti o plhjusmìc thc Ell�dac eÐnai stajerìc sta 10 e-

katommÔria, kai ton diakrÐnoume se dÔo katast�seic: touc katoÐkouc thc Attik c kai touc

katoÐkouc twn �llwn periferei¸n. Upojètoume oti k�je qrìno, to 4% tou plhjusmoÔ pou

katoikeÐ ektìc Attik c metakineÐtai sthn Attik , en¸ to 6% tou plhjusmoÔ thc Atik c

metakineÐtai se �llh perifèreia. Se aut th diadikasÐa èqoume dÔo katast�seic, �ra o Mar-

kobianìc pÐnakac ja èqei diast�seic 2 × 2. Sthn pr¸th st lh b�zoume thn kat�stash

��katoikeÐ ektìc Attik c��, kai sth deÔterh st lh thn kat�stash ��katoikeÐ sthn Attik ��.

Me ta parap�nw dedomèna èqoume ton Markobianì pÐnaka

M =

[0, 96 0, 06

0, 04 0, 94

].

Upojètoume oti to 1970 katoikoÔsan 8 ekatommÔria ektìc Attik c kai 2 ekatommÔria sthn

Attik . Dhlad èqoume arqikì di�nusma katanom c

[0, 8

0, 2

]. Sto epìmeno b ma, to 1971,

h katanom tou plhjusmoÔ ja eÐnai[0, 96 0, 06

0, 04 0, 94

][0, 8

0, 2

]=

[0, 78

0, 22

],


to 1972 ja eÐnai [0, 96 0, 06

0, 04 0, 94

]2 [0, 8

0, 2

]=

[0, 762

0, 238

],

en¸ met� apì k èth h katanom ja èqei gÐnei[xk

yk

]=

[0, 96 0, 06

0, 04 0, 94

]k [0, 8

0, 2

].

E�n o pÐnakac M eÐnai diagwniopoi simoc, mporoÔme eÔkola na upologÐsoume tic dun�meic,

kai na melet soume th makroprìjesmh katanom tou plhjusmoÔ.

Ac jewr soume ènan 2× 2 Markobianì pÐnaka,

M =

[1− a b

a 1− b

].

UpologÐzoume to qarakthristikì polu¸numo,

χM (λ) =

∣∣∣∣∣ 1− a− λ b

a 1− b− λ

∣∣∣∣∣ = (1− λ)(1− a− b− λ) .

Oi idiotimèc eÐnai λ1 = 1 kai λ2 = 1− a− b, kai ta antÐstoiqa idiodianÔsmata eÐnai

x1 =

[b

a

]kai x2 =

[−1

1

].

Jètoume R =

[b −1

a 1

]kai èqoume

Ak = R

[1 0

0 λk2

]R−1 .

Epistrèfoume sto Par�deigma 1.12, me thn arqik katanom tou plhjusmoÔ (0, 8 , 0, 2),

kai brÐskoume

Mk =

[b −1

a 1

][1 0

0 λk2

]1

a+ b

[1 1

−a b

][0, 8

0, 2

]

=1

a+ b

[b− λk2(0, 2b− 0, 8a)

a+ λk2(0, 2b− 0, 8a)

].

E�n 0 < a < 1 kai 0 < b < 1, tìte 0 < a + b < 2, kai −1 < 1 − a − b < 1. Sunep¸c

λk2 → 0 kaj¸c k → ∞. Gia meg�la k to di�nusma katanom c teÐnei sto1

a+ b

[b

a

]. Sto

par�deigma, me a = 0, 04, b = 0, 06, h katanom tou plhjusmoÔ metaxÔ thc upìloiphc q¸rac

kai thc Attik c teÐnei proc mÐa analogÐa 3 : 2. ParathroÔme oti h makroprìjesmh katanom


den exart�tai apì thn arqik katanom , oÔte apì tic akribeÐc timèc twn a kai b, all� mìnon

apì to lìgo b : a.

Den up�rqei p�nta mÐa makroprìjesmh katanom proc thn opoÐa teÐnei h Markobian

diadikasÐa. Sto 2 × 2 par�deigma, e�n a = b = 1, tìte M =

[0 1

1 0

]kai oi plhjusmoÐ

enall�ssontai se k�je b ma, [0 1

1 0

][x

y

]=

[y

x

].

MÐa Markobian diadikasÐa eÐnai kanonik (regular) e�n k�poia dÔnamh tou Markobia-

noÔ pÐnaka èqei ìla ta stoiqeÐa gn sia jetik�. Gia tètoiec diadikasÐec èqoume to akìloujo

apotèlesma

Je¸rhma 1.7 E�n M eÐnai o n × n pÐnakac mÐac kanonik c Markobian c diadikasÐac,

tìte λ1 = 1 eÐnai idiotim me pollaplìthta 1, kai k�je �llh idiotim λi, i = 2, . . . , n èqei

thn idiìthta |λi| < 1. H Markobian diadikasÐa èqei makroprìjesmh katanom , pou eÐnai

idiodi�nusma gia thn idiotim 1.

H diadikasÐa sto Par�deigma 1.12 eÐnai kanonik , afoÔ o pÐnakac M den èqei mhdeni-

k� stoiqeÐa. H makroprìjesmh katanom pou upologÐsame eÐnai pr�gmati idiodi�nusma thc

idiotim c 1.

Par�deigma 1.13 Na sumplhrwjeÐ

Sust mata grammik¸n diaforik¸n exis¸sewn

Na sumplhrwjeÐ

'Askhsh 1.14 E�n A =

[4 3

1 2

], diagwniopoi ste ton A kai upologÐste ton

pÐnaka A100.

'Askhsh 1.15 Paragontopoi ste touc akìloujouc pÐnakec sth morf A =

RDR−1, ìpou D eÐnai diag¸nioc pÐnakac.[1 2

0 3

]kai

[1 1

2 2

].


Di�lexh 5

Idiotimèc grammikoÔ telest

JewroÔme èna dianusmatikì q¸ro V (ìqi upoqrewtik� peperasmènhc di�stashc) p�nw apì

to s¸ma K. Ja melet soume grammikèc apeikonÐseic apì ton V ston eautì tou,

L : V −→ V .

MÐa tètoia apeikìnish onom�zetai grammikìc telest c sto V ( endomorfismìc tou

V ).

Orismìc 1.2. JewroÔme grammikì telest L : V −→ V . Oi arijmoÐ λ tou K gia touc

opoÐouc up�rqoun mh mhdenik� dianÔsmata v ∈ V pou ikanopoioÔn thn exÐswsh

Lv = λv (1.7)

onom�zontai idiotimèc tou grammikoÔ telest L, en¸ ta mh mhdenik� dianÔsmata pou

ikanopoioÔn thn 1.7 onom�zontai idiodianÔsmata tou L gia thn idiotim λ. To sÔnolo

twn idiodianusm�twn tou L gia thn idiotim λ, mazÐ me to di�nusma 0, apoteleÐ èna grammikì

upìqwro tou V pou onom�zetai idiìqwroc tou L gia thn idiotim λ.

Drasthriìthta 1.13 Elègxte oti pr�gmati o idiìqwroc tou L gia thn idiotim

λ eÐnai grammikìc upìqwroc tou V , dhlad oti eÐnai kleistìc wc proc tic pr�xeic tou

dianusmatikoÔ q¸rou.

Par�deigma 1.14 O telest c L = aIV : v 7→ av èqei monadik idiotim a. K�je mh

mhdenikì di�nusma tou V eÐnai idiodi�nusma tou L gia thn idiotim a. O idioq¸roc tou L gia

thn idiotim a eÐnai ìloc o q¸roc V .

Par�deigma 1.15 O telest c L(x, y) = (3x, 12y) èqei idiodi�nusma (1, 0) gia thn idio-

tim 3 kai idiodi�nusma (0, 1) gia thn idiotim 12 .

Par�deigma 1.16 O telest c peristrof c kat� π2 sto R2, R(x, y) = (−y, x) den èqei

kamÐa idiotim sto R.O telest c peristrof c kat� π

2 sto C, R(z) = iz èqei idiotim i, me idiìqwro ìlo to C.

Par�deigma 1.17 JewroÔme ton telest parag¸gishc sto dianusmatikì q¸ro twn po-

luwnÔmwn me suntelestèc sto K, D : K[x] −→ K[x], pou apeikonÐzei k�je polu¸numo

p(x) = anxn+· · ·+a1x+a0 sthn tupik par�gwgì tou, D(p(x)) = nanx

n−1+· · ·+2a2x+a1.

AfoÔ deg(D(p(x))) = deg(p(x)) − 1, kanèna polu¸numo jetikoÔ bajmoÔ den apeikonÐzetai

se pollapl�sio tou eautoÔ tou. H monadik idiotim tou telest D eÐnai λ = 0, kai èna

idiodi�nusma eÐnai to stajerì polu¸numo p(x) = 1.


Par�deigma 1.18 JewroÔme ton telest shift sto dianusmatikì q¸ro twn akolouji¸n

me ìrouc sto s¸ma K, s : KN −→ KN, pou apeikonÐzei k�je akoloujÐa (an) sthn akoloujÐa

(bn), ìpou bn = an+1. O telest c shift èqei k�je mh mhdenikì stoiqeÐo tou K wc idiotim , me

idiodi�nusma tic antÐstoiqec gewmetrikèc akoloujÐec: E�n λ 6= 0 kai xλ eÐnai h gewmetrik

akoloujÐa xλ, n = aλn, tìte s(xλ) = λxλ.

Prìtash 1.8 E�n L : V −→ V kai M : W −→ W eÐnai grammikoÐ telestèc kai up�rqei

isomorfismìc T : V −→W tètoioc ¸ste

L = T−1 ◦M ◦ T

tìte oi telestèc L kai M èqoun tic Ðdiec idiotimèc. EpÐ plèon, e�n v eÐnai idiodi�nusma tou L

gia thn idiotim λ, tìte T (v) eÐnai idiodi�nusma tou M gia thn Ðdia idiotim .

Apìdeixh. Gia k�je v ∈ V èqoume L(v) = λv e�n kai mìnon e�n T−1 ◦M ◦ T (v) = λv,

dhlad M ◦ T (v) = T (λv) = λT (v). Sunep¸c λ eÐnai idiotim tou L, me idiodi�nusma v, e�n

kai mìnon e�n λ eÐnai idiotim tou M , me idiodi�nusma T (v).

�

L mma 1.9 O arijmìc λ ∈ K eÐnai idiotim tou telest L : V −→ V e�n kai mìnon e�n

L−λIV den eÐnai monomorfismìc. Se aut n thn perÐptwsh, o idioq¸roc thc idiotim c λ eÐnai

o pur nac ker(L−λIV ), kai k�je mh mhdenikì di�nusma tou ker(L−λIV ) eÐnai idiodi�nusma

tou L gia thn idiotim λ.

Apìdeixh. E�n L − λIV den eÐnai monomorfismìc, tìte up�rqei mh mhdenikì di�nusma

v ∈ V tètoio ¸ste (L − λIV )(v) = 0, dhlad L(v) = λv, kai sunep¸c v eÐnai idiodi�nusma

tou L kai λ idiotim tou L.

AntÐstrofa, e�n λ ∈ K eÐnai idiotim tou L, tìte up�rqei mh mhdenikì di�nusma v ∈ Vtètoio ¸ste L(v) = λv, kai sunep¸c (L−λIV )(v) = 0, �ra L−λIV den eÐnai monomorfismìc.

�

Polu¸numa kai telestèc

JewroÔme èna polu¸numo p me suntelestèc sto K,

p(t) = aktk + · · ·+ a1t+ a0 .

E�n L : V −→ V eÐnai grammikìc telest c sto V , tìte oi dun�meic Li = L ◦ · · · ◦ L, ìpousunjètoume i forèc ton telest L, eÐnai epÐshc grammikoÐ telestèc sto V . MporoÔme na


antikatast soume ton telest L sth jèsh thc metablht c tou poluwnÔmou,

p(L) = akLk + · · ·+ a1L+ a0IV ,

kai to apotèlesma eÐnai p�li ènac grammikìc telest c sto V ,

p(L) : V −→ V : v 7−→ akLk(v) + · · ·+ a1L(v) + a0v .

ParathroÔme oti e�n p(x), q(x) eÐnai polu¸numa, oi telestèc p(L) kai q(L) metatÐjentai:

p(L)q(L) = (pq)(L) = (qp)(L) = q(L)p(L) .

'Uparxh idiotim¸n

Je¸rhma 1.10 K�je telest c se èna mh mhdenikì dianusmatikì q¸ro peperasmènhc

di�stashc, p�nw apì to C, èqei toul�qiston mÐa idiotim .

Apìdeixh. JewroÔme dianusmatikì q¸ro V , dimV = n, èna grammikì telest L : V −→V , kai èna mh mhdenikì di�nusma v ∈ V . Tìte h sullog v, L(v), L2(v), . . ., Ln(v) èqei

n + 1 stoiqeÐa, kai sunep¸c eÐnai grammik� exarthmènh. Dhlad up�rqoun arijmoÐ ai ∈ C,ìqi ìloi mhdèn, tètoioi ¸ste

a0v + a1L(v) + · · ·+ anLn(v) = 0 .

E�n a0 = 0, tìte k�poio apì ta ai gia i ≥ 1 eÐnai diaforetikì apì mhdèn. E�n a0 6= 0, afoÔ

v 6= 0, p�li k�poio apì ta ai gia i ≥ 1 eÐnai diaforetikì apì mhdèn. Sunep¸c to polu¸numo

p(t) = a0 + a1t+ · · ·+ antn èqei bajmì k, gia 1 ≤ k ≤ n. Dhlad up�rqei jetikìc akèraioc

k ≤ n, tètoioc ¸ste ak 6= 0 kai ai = 0 gia k�je i = k+1, . . . , n. SÔmfwna me to Jemeli¸dec

Je¸rhma thc 'Algebrac, (Shmei¸seic EpÐpedo kai Q¸roc, Kef�laio 3) to polu¸numo p(x)

paragontopoieÐtai se ginìmeno k diwnÔmwn, dhlad up�rqoun migadikoÐ arijmoÐ c1, . . . , ck

tètoioi ¸ste

p(x) = ak(x− c1) · · · (x− ck) .

Sunep¸c o telest c p(L) eÐnai Ðsoc me ton telest ak(L− c1IV ) ◦ · · · ◦ (L− ckIV ), kai

ak(L− c1IV ) ◦ · · · ◦ (L− ckIV )(v) = 0 .

AfoÔ v 6= 0, o telest c (L− c1IV ) ◦ · · · ◦ (L− ckIV ) den eÐnai monomorfismìc, kai sumperaÐ-

noume oti up�rqei toul�qiston èna i, 1 ≤ i ≤ k, gia to opoÐo h apeikìnish L− ciIV den eÐnai

eneikonik . Sunep¸c up�rqei mh mhdenikì w ∈ V tètoio ¸ste (L − ciIV )(w) = 0, dhlad

L(w) = ciw, kai λ = ci eÐnai idiotim tou telest L.

�


Telestèc kai pÐnakec

JewroÔme dianusmatikì q¸ro V peperasmènhc di�stashc n kai b�sh B = {x1, . . . , xn} touV . E�n L : V −→ V eÐnai ènac grammikìc telest c, tìte o pÐnakac BLB pou parist�nei ton

L wc proc th b�sh B eÐnai o pÐnakac pou èqei sth gramm i kai st lh j to stoiqeÐo aij pou

orÐzetai apì th sqèsh

L(xj) =

n∑i=1

aijxi .

Prìtash 1.11 JewroÔme dianusmatikì q¸ro peperasmènhc di�stashc, b�sh B tou V ,

kai grammikì telest L : V −→ V . Tìte λ ∈ K eÐnai idiotim tou telest L e�n kai mìnon

e�n λ eÐnai idiotim tou pÐnaka BLB pou parist�nei ton L wc proc th b�sh B.

Apìdeixh. E�n vB sumbolÐzei to di�nusma suntetagmènwn tou dianÔsmatoc v ∈ V wc proc

th b�sh B, tìte, apì ton orismì tou BLB èqoume

(L(v))B = BLB vB .

Sunep¸c L(v) = λv e�n kai mìnon e�n BLB vB = (λv)B = λvB.

�

DÔo pÐnakec A kai B lègontai ìmoioi e�n up�rqei antistrèyimoc pÐnakac S tètoioc ¸ste

A = S−1BS. Apì thn Prìtash 1.8 kai thn Prìtash 1.11 èqoume to akìloujo sumpèrasma.

Prìtash 1.12 DÔo ìmoioi pÐnakec èqoun tic Ðdiec idiotimèc.

�

E�n V eÐnai dianusmatikìc q¸roc peperasmènhc di�stashc kai L eÐnai ènac grammikìc

telest c ston V , jewroÔme pÐnakec A kai B pou antistoiqoÔn ston L wc proc diaforetikèc

b�seic tou V . GnwrÐzoume apì thn Eisagwg sth Grammik 'Algebra oti oi pÐnakec A kai

B eÐnai ìmoioi, kai sunep¸c oti detA = detB. EÔkola blèpoume oti oi pÐnakec poluwnÔmwn

A− λIn kai B− λIn eÐnai epÐshc ìmoioi, kai sunep¸c ta qarakthristik� polu¸numa twn dÔo

pin�kwn eÐnai Ðsa,

det(A− λIn) = det(B − λIn) .

SumperaÐnoume oti mporoÔme na orÐsoume to qarakthristikì polu¸numo χL(λ) tou

telest L, na eÐnai to qarakthristikì polu¸numo tou pÐnaka tou L wc proc opoiad pote

b�sh tou V .

H akìloujh prìtash eÐnai sunèpeia tou antÐstoiqou apotelèsmatoc gia pÐnakec.

Prìtash 1.13 JewroÔme telest L : V −→ V , kai λ1, . . . , λm diaforetikèc idiotimèc

tou L, me antÐstoiqa idiodianÔsmata v1, . . . , vm. Tìte to sÔnolo {v1, . . . , vm} eÐnai grammik�anex�rthto.


Pìrisma 1.14 K�je telest c sto dianusmatikì q¸ro peperasmènhc di�stashc V , èqei

to polÔ dimV diaforetikèc idiotimèc.

�


Di�lexh 6

AnalloÐwtoi upìqwroi

JewroÔme ènan grammikì telest L : V −→ V se èna dianusmatikì q¸ro V . GnwrÐzoume

oti ta sÔnola kerL kai imL eÐnai upìqwroi tou V , kai eÔkola elègqoume oti

L(kerL) ⊆ kerL

L(imL) ⊆ imL .

O idiìqwroc Xλ mÐac idiotim c tou L eÐnai epÐshc ènac upìqwroc tou V me thn idiìthta

L(Xλ) ⊆ Xλ.

Orismìc 1.3. L : V −→ V grammikìc telest c. O upìqwroc X ⊆ V onom�zetai

analloÐwtoc upìqwroc apì ton telest L, e�n

L(X) ⊆ X .

Prosèxte oti den upojètoume oti L(X) = X, oÔte oti L−1(X) ⊆ X.

Par�deigma 1.19 Gia k�je telest L : V −→ V , oi upìqwroi {0}, V , kerL kai imL

eÐnai analloÐwtoi upìqwroi tou L.

Par�deigma 1.20 Gia k�je telest L : V −→ V , kai k�je idiotim λ tou L, oi upìqwroi

tou idiìqwrou thc λ eÐnai analloÐwtoi upìqwroi tou L.

Par�deigma 1.21 JewroÔme ton telest L(x, y, z) = (x + z, y, z − y). O upìqwroc

U = {(x, y, z) ∈ K3 : y = 0} eÐnai analloÐwtoc apì ton telest L: L(x, 0, z) = (x +

z, 0, z) ∈ U . O upìqwroc W = {(x, y, z) ∈ K3 : x = y = 0} den eÐnai analloÐwtoc apì ton

telest L: L(0, 0, 1) = (1, 0, 1) 6∈W .

Par�deigma 1.22 JewroÔme ton telest parag¸gishc sto dianusmatikì q¸ro twn po-

luwnÔmwn me suntelestèc sto K, D : K[x] −→ K[x], kai ton upìqwro Km[x]twn poluw-

nÔmwn bajmoÔ mikrìterou Ðsou me m. AfoÔ k�je polu¸numo bajmoÔ k apeikonÐzetai se

polu¸numo mikrìterou bajmoÔ, o upìqwroc Km[x] eÐnai analloÐwtoc apì ton telest D.

Par�deigma 1.23 JewroÔme ton telest shift sto dianusmatikì q¸ro twn akolouji¸n

me ìrouc sto s¸ma K, s : KN −→ KN, kai ton upìqwro X twn akolouji¸n pou eÐnai

fragmènec. AfoÔ mÐa fragmènh akoloujÐa paramènei fragmènh ìtan ��xeq�soume�� ton pr¸to

ìro thc, o upìqwroc X eÐnai analloÐwtoc apì ton telest shift.

Prìtash 1.15 JewroÔme dianusmatikì q¸ro V kai grammikì telest L : V −→ V .

E�n X kai Y eÐnai upìqwroi tou V analloÐwtoi apì ton L, tìte X ∩ Y kai X + Y eÐnai

upìqwroi analloÐwtoi apì ton L.


Prìtash 1.16 E�n L : V −→ V eÐnai telest c sto q¸ro V , kai X eÐnai upìqwroc tou

V analloÐwtoc apì ton L, me dimV = n kai dimX = k, tìte mporoÔme na epilèxoume

kat�llhlh b�sh tou V ètsi ¸ste o pÐnakac tou L wc proc thn epilegmènh b�sh na eÐnai

thc morf c [A B

0 C

]me èna (n− k)× k mhdenikì mplìk k�tw arister�.

Apìdeixh. GnwrÐzoume oti k�je b�sh {x1, . . . , xk} tou X mporeÐ na epektajeÐ se b�sh

{x1, . . . , xk, vk+1, . . . , vn} tou V . E�n X eÐnai analloÐwtoc apì ton telest L : V −→ V ,

kai [aij ] eÐnai o pÐnakac tou L wc proc th b�sh {x1, . . . , xk, vk+1, . . . , vn} tìte

L(xj) =k∑i=1

aijxi +n∑

i=k+1

aijvi .

'Omwc L(xj) ∈ X kai sunep¸c aij = 0 gia i = k + 1, . . . , n.

'Ara o pÐnakac (aij) èqei èna (n− k)× k mhdenikì mplìk k�tw arister�.

�

Apì tic idiìthtec thc orÐzousac sthn Eisagwg sth Grammik 'Algebra, gnwrÐzoume oti

gia èna pÐnaka thc morf c

A =

[B F

0 C

](1.8)

isqÔei detA = detB detC. Efarmìzontac aut thn idiìthta ston pÐnaka poluwnÔmwn A−λI,

èqoume thn akìloujh Prìtash4.

Prìtash 1.17 JewroÔme dianusmatikì q¸ro V kai grammikì telest L : V −→ V .

E�n X upìqwroc tou V analloÐwtoc apì ton L, tìte to qarakthristikì polu¸numo tou L|XdiaireÐ to qarakthristikì polu¸numo tou L,

χL|X (λ) |χL(λ) .

'Otan o dianusmatikìc q¸roc V diasp�tai se eujÔ �jroisma dÔo perissìterwn upìsw-

rwn, k�je ènac ek twn opoÐwn eÐnai analloÐwtoc apì ènan telest , tìte kai o pÐnakac mporeÐ

na p�rei th morf diag¸niou pÐnaka se mplok. Sthn epìmenh prìtash apodeiknÔoume to

apotèlesma gia dÔo upìqwrouc, kai af noume wc �skhsh th genÐkeush, me qr sh epagwg c.

4O sumbolismìc a | b shmaÐnei oti to a diaireÐ to b.


Prìtash 1.18 JewroÔme dianusmatikì q¸ro V peperasmènhc di�stashc n, kai upìqw-

rouc X kai Y tou V , tètoiouc ¸ste

V = X ⊕ Y ,

me b�seic B = {x1, . . . , xk} tou X kai C = {yk+1, . . . , yn} tou Y . E�n L : V −→ V eÐnai

grammikìc telest c kai X, Y eÐnai analloÐwtoi upìqwroi tou L, tìte o pÐnakac tou L wc

proc th b�sh S = {x1, . . . , xk, yk+1, . . . , yn} eÐnai diag¸nioc se mplok, dhlad èqei th

morf

SLS =

[B 0

0 C

],

ìpou B eÐnai o k×k pÐnakac tou telest L|X wc proc th b�sh B kai C eÐnai o (n−k)×(n−k)

pÐnakac tou telest L|Y wc proc th b�sh C.Tìte gia to qarakthristikì polu¸numo tou L isqÔei

χL(λ) = χL|X (λ)χL|Y (λ) .

Apìdeixh. E�n o pÐnakac tou L wc proc th b�sh S eÐnai [aij ], tìte gia j = 1, . . . , k,

L(xj) =

k∑i=1

aijxi +

n∑i=k+1

aijyi .

All� L(xj) ∈ X, �ra aij = 0 gia i = k + 1, . . . , n.

Parìmoia, gia j = k + 1, . . . , n, aij = 0 gia i = 1, . . . , k.

H idiìthta twn qarakthristik¸n poluwnÔmwn eÐnai sunèpeia thc 1.8.

�

B�seic apì idiodianÔsmata

Prìtash 1.19 E�n L : V −→ V eÐnai grammikìc telest c, kai o dianusmatikìc q¸roc

V èqei mÐa peperasmènh b�sh apì idiodianÔsmata tou L, tìte o pÐnakac tou L wc proc aut n

th b�sh eÐnai diag¸nioc, me tic idiotimèc tou telest sth diag¸nio.

Apìdeixh. 'Estw {v1, . . . , vn} mÐa b�sh apì idiodianÔsmata, kai λ1, . . . λn oi antÐstoiqec

idiotimèc. Tìte

L(vj) =n∑i=1

aijvi = λjvj .

AfoÔ ta v1, . . . , vn eÐnai grammik� anex�rthta, oi suntelestèc eÐnai monadikoÐ, kai

aij =

{0 e�n i 6= j

λj e�n i = j .


Sunep¸c o pÐnakac [aij ] eÐnai diag¸nioc.

�

EÐnai profanèc oti isqÔei kai to antÐstrofo: e�n o pÐnakac tou telest L wc proc k�poia

b�sh eÐnai diag¸nioc, tìte ta stoiqeÐa thc b�shc eÐnai idiodianÔsmata tou L.

E�n dimV = n kai o L èqei n diaforetikèc idiotimèc, tìte up�rqei mÐa b�sh tou V h

opoÐa apoteleÐtai apì idiodianÔsmata tou L, kai wc proc thn opoÐa o pÐnakac tou L eÐnai dia-

g¸nioc. 'Eqoume dei ìmwc paradeÐgmata ìpou up�rqoun ligìtera apì n grammik� anex�rthta

idiodianÔsmata, kai den up�rqei b�sh wc proc thn opoÐa o pÐnakac tou L eÐnai diag¸nioc.

UpenjumÐzoume oti h gewmetrik pollaplìthta mÐac idiotim c eÐnai h di�stash tou idiì-

qwrou thc idiotim c.

Prìtash 1.20 H gewmetrik pollaplìthta mÐac idiotim c eÐnai Ðsh mikrìterh apì thn

algebrik pollaplìthta.

Apìdeixh. Upojètoume oti h idiotim λ1 èqei gewmetrik pollaplìthta k, kai epilègoume

grammik� anex�rthta idiodianÔsmata thc idiotim c λ1, v1, . . . , vk. Sumplhr¸noume to sÔnolo

{v1, . . . , vk} se b�sh B = {v1, . . . , vk, vk+1, . . . , vn} tou V . AfoÔ gia j = 1, . . . , k,

L(vj) = λ1vj , h st lh j tou pÐnaka A =B LB èqei stoiqeÐa ajj = λ1 kai aij = 0 gia i 6= j.

Sunep¸c to qarakthristikì polu¸numo χL(λ) diaireÐtai apì to (λ−λ1)k. 'Ara k den mporeÐ

na eÐnai megalÔtero apì thn pollaplìthta thc rÐzac λ1 sto χL.

�


Di�lexh 7

TrigwnikoÐ pÐnakec

UpenjumÐzoume oti ènac n × n pÐnakac eÐnai �nw trigwnikìc ìtan èqei mhdenik� se ìlec tic

jèseic k�tw apì th diag¸nio, dhlad ìtan aij = 0 gia k�je i > j. 'Enac �nw trigwnikìc

pÐnakac eÐnai idiìmorfoc e�n kai mìnon e�n èqei mhdenikì stoiqeÐo sth diag¸nio. Se aut thn

par�grafo ja doÔme oti p�nw apì to C mporoÔme p�nta na broÔme mÐa b�sh tou V wc proc

thn opoÐa o pÐnakac tou L eÐnai �nw trigwnikìc.

Prìtash 1.21 JewroÔme grammikì telest L : V −→ V , kai b�sh B = {v1, . . . , vn}tou V . Ta akìlouja eÐnai isodÔnama:

1. O pÐnakac A tou L wc proc th b�sh B eÐnai �nw trigwnikìc

2. L(vj) ∈ 〈v1, . . . , vj〉 gia j = 1, . . . , n.

3. Gia k�je j = 1, . . . , n o upìqwroc 〈v1, . . . , vj〉 eÐnai analloÐwtoc apì ton L.

Apìdeixh. To 2 shmaÐnei oti to di�nusma suntetagmènwn tou L(vj) wc proc th b�sh Bèqei mhdenik� stic teleutaÐec n− j jèseic, pou eÐnai akrib¸c to Ðdio me to 1. EÐnai profanèc

oti to 3 sunep�getai to 2. Ja deÐxoume oti to 2 sunep�getai to 3. E�n v ∈ 〈v1, . . . , vj〉, tìtev = a1v1 + · · ·+ ajvj . E�n isqÔei to 2, gia k�je i = 1, . . . , j

L(vi) ∈ 〈v1, . . . , vi〉 ⊆ 〈v1, . . . , vj〉 .

Sunep¸c

L(v) = a1L(v1) + · · ·+ ajL(vj) ∈ 〈v1, . . . , vj〉 .

�

Orismìc 1.4. 'Enac n × n pÐnakac A me stoiqeÐa sto K eÐnai trigwnopoi simoc

(p�nw apì to K) e�n eÐnai ìmoioc me ènan �nw trigwnikì pÐnaka U , dhlad e�n up�rqei

antistrèyimoc pÐnakac R me stoiqeÐa sto K tètoioc ¸ste A = RUR−1.

'Enac telest c L : V −→ V eÐnai trigwnopoi simoc e�n up�rqei mÐa diatetagmènh b�sh

tou V wc proc thn opoÐa o pÐnakac tou L eÐnai �nw trigwnikìc.

Prìtash 1.22 Upojètoume oti o telest c L : V −→ V eÐnai trigwnopoi simoc. Tìte oi

idiotimèc tou L eÐnai akrib¸c ta stoiqeÐa thc diagwnÐou tou trigwnikoÔ pÐnaka pou parist�nei

ton L.


Apìdeixh. JewroÔme ton �nw trigwnikì pÐnaka A thc L, wc proc th b�sh B:

A =

λ1 ∗

. . .

0 λn

Tìte h apeikìnish L− λIV , gia λ ∈ K, èqei pÐnaka wc proc th b�sh B:

λ1 − λ ∗. . .

0 λn − λ

o opoÐoc eÐnai idiìmorfoc e�n kai mìnon e�n λ eÐnai Ðso me k�poio apì ta stoiqeÐa thc diagw-

nÐou, λ1, . . . , λn.

'Ara oi idiotimèc tou L eÐnai akrib¸c λ1, . . . , λn.

�

Je¸rhma 1.23 JewroÔme dianusmatikì q¸ro V peperasmènhc di�stashc p�nw apì to

C, kai grammikì telest L : V → V . Tìte o telest c L eÐnai trigwnopoi simoc.

Apìdeixh. Ja qrhsimopoi soume epagwg sth di�stash tou V . E�n dimV = 1, arkeÐ

na parathr soume oti k�je 1× 1 pÐnakac eÐnai �nw trigwnikìc.

Upojètoume oti dimV = n ≥ 2. AfoÔ briskìmaste p�nw apì touc migadikoÔc arijmoÔc,

apì to Je¸rhma 1.10, o L èqei toul�qiston mÐa idiotim λ1. 'Estw u1 èna idiodi�nusma gia

thn idiotim λ1. Sumplhr¸noume to {u1} se b�sh B = {u1, u2, . . . , un} tou V , kai jewroÔmeton pÐnaka tou L wc proc th b�sh B, A = BLB. H pr¸th st lh tou A perièqei to di�nusma

suntetagmènwn tou L(u1) = λ1u1 wc proc th b�sh B. Sunep¸c o A èqei th morf

A =

λ1 a12 . . . a1n

0... D

0

.

JewroÔme to V wc eujÔ �jroisma twn upìqwrwn V1 = 〈u1〉 kai U = 〈u2, . . . , un〉, ètsi¸ste k�je v ∈ V gr�fetai me monadikì trìpo wc �jroisma v = a1u1 + u, gia a1 ∈ C kai

u ∈ U . 'Eqoume apeikonÐseic j : U −→ V : u 7→ u kai p : V −→ U : v 7→ u, kai orÐzoume

M = p ◦ L ◦ j : U −→ U .

Gia i = 2, . . . , n, M(ui) = a2iu2 + · · · + aniun, ìpou (a2i, . . . , ani) eÐnai h st lh tou

pÐnaka D pou antistoiqeÐ sthn i st lh tou pÐnaka A. Sunep¸c D eÐnai o pÐnakac pou

parist�nei thn apeikìnish M wc proc th b�sh {u2, . . . , un}. AfoÔ dimU = n − 1, apì

thn epagwgik upìjesh, up�rqei b�sh W = {w2, . . . , wn}, wc proc thn opoÐa o pÐnakac thc


apeikìnishc M eÐnai �nw trigwnikìc. Exet�zoume t¸ra ton pÐnaka tou L wc proc th b�sh

B′ = {u1, w2, . . . , wn}. Autìc èqei th morf

B =

λ1 b12 . . . b1n

0... T

0

,

ìpou T eÐnai o (n− 1)× (n− 1) pÐnakac o opoÐoc parist�nei thn apeikìnish M wc proc th

b�sh W. Sunep¸c o T eÐnai �nw trigwnikìc. SumperaÐnoume oti o pÐnakac B tou telest L

wc proc th b�sh B′ eÐnai �nw trigwnikìc.

�

Prìtash 1.24 K�je tetragwnikìc pÐnakac me stoiqeÐa sto C eÐnai trigwnopoi simoc.

Par�deigma 1.24 Sto q¸ro C3[x] twn poluwnÔmwn bajmoÔ Ðsou mikrìterou apì 3, me

thn kanonik diatetagmènh b�sh B = {x3, x2, x, 1}, o telest c parag¸gishc D : C3[x] −→C3[x] parist�netai apì ton pÐnaka

BDB =

0 0 0 0

3 0 0 0

0 2 0 0

0 0 1 0

.

Blèpoume oti o pÐnakac eÐnai k�tw trigwnikìc. Gia na ton metatrèyoume se �nw trigwnikì

arkeÐ mÐa anadi�taxh thc b�shc, F = {1, x, x2, x3}. O pÐnakac met�bashc apì th b�sh Bsth b�sh F eÐnai

R =

0 0 0 1

0 0 1 0

0 1 0 0

1 0 0 0

,

me R−1 = R. 'Ara o �nw trigwnikìc pÐnakac eÐnai

FDF = R BDBR =

0 1 0 0

0 0 2 0

0 0 0 3

0 0 0 0

.

Ac efarmìsoume th diadikasÐa thc apìdeixhc tou Jewr matoc 1.23 gia na trigwnopoi-

soume ton telest D. O D èqei monadik idiotim λ = 0, me idiodi�nusma to stajerì

polu¸numo 1. Epilègoume wc pr¸to stoiqeÐo thc b�shc autì to polu¸numo, kai jewroÔme


th diatetagmènh b�sh C = {1, x3, x2, x} wc proc thn opoÐa o pÐnakac tou telest D eÐnai0 0 0 1

0 0 0 0

0 3 0 0

0 0 2 0

.O k�tw dexi� 3× 3 pÐnakac

B =

0 0 0

3 0 0

0 2 0

parist�nei ton telest M : 〈x3, x2, x〉 −→ 〈x3, x2, x〉 pou apeikonÐzei ta x3, x2, x sta

3x2, 2x, 0 antÐstoiqa. Autìc o telest c èqei trigwnikì pÐnaka wc proc th b�sh C′ =

{x, x2, x3}. Apì aut th diadikasÐa, katal goume oti wc proc th b�sh {1, x, x2, x3} = Fo pÐnakac tou telest D eÐnai �nw trigwnikìc, pou epalhjeÔei to prohgoÔmeno apotèlesma.

'Otan o dianusmatikìc q¸roc V eÐnai p�nw apì to s¸ma twn pragmatik¸n arijm¸n, den

eÐnai dedomènh h Ôparxh idiotim¸n. Se aut thn perÐptwsh gia na exasfalÐsoume thn Ôparxh

b�shc wc proc thn opoÐa o telest c eÐnai �nw trigwnikìc, prèpei na upojèsoume oti to

qarakthristikì polu¸numo tou telest eÐnai ginìmeno paragìntwn bajmoÔ 1. Diatup¸noume

to apotèlesma sthn perÐptwsh enìc n× n pÐnaka.

Je¸rhma 1.25 JewroÔme n×n pÐnaka A me stoiqeÐa sto K. Tìte o A eÐnai trigwnopoi-

simoc e�n kai mìnon e�n to qarakthristikì polu¸numo χA(λ) eÐnai ginìmeno paragìntwn

bajmoÔ 1 p�nw apì to K.

Apìdeixh. E�n o A eÐnai trigwnopoi simoc, jewroÔme trigwnikì pÐnaka B ìmoio me ton A,

me stoiqeÐa λ1, . . . , λn sth diag¸nio. Tìte det(A−λI) = det(B−λI) = (λ1−λ) · · · (λn−λ),

kai sunep¸c χA(λ) eÐnai ginìmeno paragìntwn bajmoÔ 1 p�nw apì to K.

E�n to qarakthristikì polu¸numo tou A eÐnai ginìmeno prwtob�jmiwn paragìntwn p�nw

apì to K, ìpwc kai sthn apìdeixh tou Jewr matoc 1.23 ja qrhsimopoi soume epagwg sto

n. 'Estw λ1 tètoio ¸ste (λ1−λ) diaireÐ to χA(λ). Tìte λ1 eÐnai idiotim tou A; èstw u1 èna

idiodi�nusma thc λ1. JewroÔme antistrèyimo pÐnaka R me pr¸th st lh u1. Tìte R−1u1 eÐnai

h pr¸th st lh tou tautotikoÔ pÐnaka In, kai R−1AR èqei pr¸th st lh R−1Au1 = λ1R−1u1.

Sunep¸c

R−1AR =

λ1 b2 . . . bn

0... B

0

,ìpou b = (b2, . . . , bn) ∈ Kn−1 kai B eÐnai (n− 1)× (n− 1) pÐnakac. Apì thn Prìtash 1.8,

χA(λ) = (λ1−λ)χB(λ), kai to qarakthristikì polu¸numo tou B eÐnai ginìmeno paragìntwn


bajmoÔ 1. Apì thn epagwgik upìjesh, o pÐnakac B eÐnai trigwnopoi simoc. 'Ara up�rqei

(n − 1) × (n − 1) antistrèyimoc pÐnakac S tètoioc ¸ste S−1BS na eÐnai �nw trigwnikìc.

Jètoume U =

[1 0

0 S

]kai èqoume

U−1R−1ARU =

[1 0

0 S

]−1 [λ1 b

0 B

][1 0

0 S

]

=

[λ1 bS

0 S−1BS

],

pou eÐnai �nw trigwnikìc.

�


Di�lexh 8

Je¸rhma Cayley – Hamilton

Par�deigma 1.25 Prin diatup¸soume to Je¸rhma Cayley – Hamilton ja upologÐsoume

to qarakthristikì polu¸numo enìc pÐnaka, ton opoÐo ja qrhsimopoi soume sthn apìdeixh.

JewroÔme ton pÐnaka

B =

0 0 . . . 0 0 a1

1 0. . . 0 0 a2

0 1. . . 0 0 a3

.... . .

. . .. . .

......

0 . . . 0 1 0 ak−1

0 . . . . . . 0 1 ak

,

dhlad tou k × k pÐnaka [bij ], me bij = 0 ìtan j 6= k kai i 6= j + 1, b(j+1) j = 1 gia

j = 1, . . . , k − 1 kai bik = ai. Autì eÐnai Ðso me thn orÐzousa

det(B − xIk) =

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

−x 0 . . . 0 0 a1

1 −x . . . 0 0 a2

0 1. . . 0 0 a3

.... . .

. . .. . .

......

0 . . . 0 1 −x ak−1

0 . . . . . . 0 1 ak − x

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣.

Gia na upologÐsoume thn orÐzousa det(B − xIn) ja qrhsimopoi soume apaloif apì k�tw

proc ta ep�nw, gia na apaleÐyoume ta −x sth diag¸nio. Afair¸ntac −x forèc thn teleutaÐa

gramm apì thn proteleutaÐa, èqoume

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

−x 0 . . . 0 0 a1

1 −x . . . 0 0 a2

0 1. . . 0 0 a3

.... . .

. . .. . .

......

0 . . . 0 1 0 ak−1 + akx− x2

0 . . . . . . 0 1 ak − x

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣.


SuneqÐzoume, afair¸ntac −x forèc th gramm i apì th gramm i − 1, gia i = k − 1, k −2, . . . , 2, kai katal goume me thn orÐzousa∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

0 0 . . . 0 0 a1 + a2x+ · · ·+ akxk−1 − xk

1 0. . . 0 0 a2 + a3x+ · · ·+ akx

k−2 − xk−1

0 1. . . 0 0

......

. . .. . .

. . ....

...

0 . . . 0 1 0 ak−1 + akx− x2

0 . . . . . . 0 1 ak − x

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣,

thn opoÐa anaptÔssoume wc proc thn pr¸th gramm kai brÐskoume to qarakthristikì po-

lu¸numo

χB(x) = det(B − xI)

= (−1)k+1(a1 + a2x+ · · ·+ akxk−1 − xk)

= (−1)k(xk − akxk−1 − · · · − a2x− a1) .

Je¸rhma 1.26 (Cayley – Hamilton) E�n χL(x) = bnxn + · · ·+ b1x+ b0 eÐnai to qa-

rakthristikì polu¸numo tou telest L, se èna dianusmatikì q¸ro peperasmènhc di�stashc

V , tìte o telest c

χL(L) = bnLn + · · ·+ b1L+ b0IV

eÐnai o mhdenikìc telest c: gia k�je v ∈ V , χL(L)(v) = 0.

Apìdeixh. Upojètoume oti χL(x) = bnxn + · · · + b1x + b0. Ja deÐxoume oti o telest c

χL(L) = bnLn + · · ·+ b1L+ b0IV paÐrnei thn tim 0 se k�je v ∈ V , kai sunep¸c oti eÐnai o

mhdenikìc telest c.

E�n v = 0 tìte profan¸c χL(L)(v) = 0. Upojètoume oti v 6= 0. E�n dimV = n,

jewroÔme th sullog twn dianusm�twn

v1 = v, v2 = L(v), v3 = L2(v), . . . , vn+1 = Ln(v) .

AfoÔ aut perièqei n + 1 dianÔsmata, eÐnai grammik� exarthmènh. Apì thn Eisagwg sth

Grammik 'Algebra gnwrÐzoume oti up�rqei jetikìc akèraioc k tètoioc ¸ste h sullog

v1, . . . , vk eÐnai grammik� anex�rthth, en¸ h sullog v1, . . . , vk+1 eÐnai grammik� exarthmènh

kai up�rqoun a0, . . . , ak ∈ K tètoia ¸ste

vk+1 = a1v1 + · · ·+ akvk .

EpekteÐnoume to grammik� anex�rthto sÔnolo {v1, . . . , vk} se b�sh tou V ,

B = {v1, . . . , vk, wk+1, . . . , wn} .


ParathroÔme oti

L(v1) = v2

L(v2) = v3...

L(vk−1) = vk

L(vk) = vk+1 = a1v1 + · · ·+ akvk

Autì shmaÐnei oti o upìqwroc 〈v1, . . . , vk〉 eÐnai analloÐwtoc apì ton L kai sunep¸c oti

o pÐnakac tou L wc proc th b�sh B èqei th morf

BLB =

[A B

0 D

].

O k× k pÐnakac A èqei sth st lh j tic k pr¸tec suntetagmènec tou L(vj) wc proc th b�sh

B. E�n j = 1, . . . , k − 1, L(vj) = vj+1, �ra a(j+1)j = 1, kai aij = 0 gia i 6= j + 1. Dhlad

o pÐnakac èqei th morf

A =

0 0 . . . 0 0 a1

1 0. . . 0 0 a2

0 1. . . 0 0 a3

.... . .

. . .. . .

......

0 . . . 0 1 0 ak−1

0 . . . . . . 0 1 ak

.

GnwrÐzoume oti

χL(x) = det(BLB − xIn)

= det(A− xIk) det(D − xIn−k)

= χA(x)χD(x)

en¸ apì to Par�deigma 1.25 èqoume oti

χA(x) = (−1)k(xk − akxk−1 − · · · − a2x− a1) .

AntikajistoÔme L gia to x kai upologÐzoume thn tim tou telest χA(L) sto v:

χA(L)(v) = (−1)k(Lk(v)− akLk−1(v)− · · · − a2L(v)− a1v)

= (−1)k(vk+1 − akvk − · · · − a2v2 − a1v1)

= 0 .

AfoÔ oi telestèc χA(L) kai χD(L) metatÐjentai,

χL(L)(v) = χD(L)χA(L)(v) = 0 .


Tèloc, afoÔ autì isqÔei gia k�je v ∈ V , χL(L) eÐnai o mhdenikìc telest c.

�

Par�deigma 1.26 JewroÔme ton telest L(x, y) = (x + 2y, 3x + 2y). O pÐnakac tou

L wc proc thn kanonik b�sh tou R2 eÐnai A =

[1 2

3 2

], kai to qarakthristikì polu¸numo

χL(λ) = χA(λ) =

∣∣∣∣∣ 1− λ 2

3 2− λ

∣∣∣∣∣ = λ2 − 3λ− 4 .

SÔmfwna me to Je¸rhma, o telest c χL(L) eÐnai o mhdenikìc telest c kai o pÐnakac

χA(A) = 0. Pr�gmati

χA(A) = A2 − 3A− 4I2 =

[7 6

9 10

]−

[3 6

9 6

]−

[4 0

0 4

]= 0 .

To Je¸rhma Cayley - Hamilton epitrèpei na aplopoioÔme parast�seic me pÐnakec, na

ekfr�zoume ton antÐstrofo enìc pÐnaka wc polu¸numo. AfoÔ A2 = 3A+ 4I2,

A3 = 3A2 + 4A = 3(3A+ 4I2) + 4A = 13A+ 12I2 ,

A4 = 13A2 + 12A = 13(3A+ 4I2) + 12A = 51A+ 52I2 , k.o.k.

AfoÔ o stajerìc ìroc tou qarakthristikoÔ poluwnÔmou χA(x) den eÐnai mhdèn, to mhdèn

den eÐnai idiotim tou pÐnaka A, kai o A eÐnai antistrèyimoc. AfoÔ A2 − 3A = 4I2, èqoume

A− 3I2 = 4A−1 kai sunep¸c A−1 = 14A−

34 I2.


A =

2 0 1

1 2 0

0 3 1

,pou èqei qarakthristikì polu¸numo χA(λ) = −λ3 + 5λ2− 8λ+ 7. Apì to Je¸rhma Cayley

– Hamilton upologÐzoume

A3 = 5A2 − 8A+ 7I3 ,

A4 = AA3 = 5A3 − 8A2 + 7A

= 5(5A2 − 8A+ 7I3)− 8A2 + 7A

= 17A2 − 33A+ 35I3 ,

A5 = 5A4 − 8A3 + 7A2

= 5(17A2 − 33A+ 35I3)− 8(5A2 − 8A+ 7I3) + 7A2 .

Me aut th diadikasÐa mporoÔme na upologÐsoume kai arnhtikèc dun�meic enìc pÐnaka

e�n autìc eÐnai antistrèyimoc. AfoÔ o stajerìc ìroc tou qarakthristikoÔ poluwnÔmou


χA(x) den eÐnai mhdèn, mporoÔme na ekfr�soume to A−1 wc polu¸numo tou A: èqoume

A2 = A3A−1 = (5A2 − 8A+ 7I3)A−1 = 5A− 8I3 + 7A−1. 'Ara

A−1 =1

7(A2 − 5A+ 8I3) .

JewroÔme to polu¸numo p(x) = x8 − 17x6 + 33x5 − 36x4 + 5x3 − 6x2 + 7x + 1. To

diairoÔme me to qarakthristikì polu¸numo χA(x):

p(x) = χA(x)(−x5 − 5x4 + x) + 2x2 + 1 .

AfoÔ χA(A) = 0, èqoume p(A) = 2A2 + I3, dhlad

p(A) = 2

2 0 1

1 2 0

0 3 1

2

+

1 0 0

0 1 0

0 0 1

=

9 6 6

8 11 2

6 18 5

.

'Askhsh 1.16 DÐdetai grammikìc telest c L : V → V , kai X, Y grammikoÐ

upìqwroi tou V analloÐwtoi apì ton L. Na exet�sete e�n oi grammikoÐ upìqwroi

X + Y kai X ∩ Y eÐnai analloÐwtoi.

'Askhsh 1.17 ApodeÐxte oti mÐa grammik apeikìnish L : V → V eÐnai eneikonik

e�n kai mìnon e�n to mhdèn den eÐnai idiìtimh thc L.

'Askhsh 1.18 BreÐte tic idiotimèc kai ta idiodianÔsmata tou telest u 7→ Au,

ìpou

A =

4 0 3

0 2 0

−3 0 4

1. sto R3

2. sto C3

'Askhsh 1.19 BreÐte ìlec tic idiotimèc kai ta idiodianÔsmata tou telest shift

sto R∞,s(a1, a2, a3, . . .) = (a2, a3, . . .)


'Askhsh 1.20 DÐdontai oi pÐnakec

A1 =

1 −2 2

−2 1 2

−2 0 3

kai

A2 =

15 7 −7

−1 1 1

13 7 −5

1. Gia k�je pÐnaka, breÐte to qarakthristikì polu¸numo kai tic idiotimèc.

2. Gia k�je idiotim , breÐte èna idiodi�nusma, kai ton idiìqwro.

3. E�n qrei�zetai sumplhr¸sete èna grammik� anex�rthto sÔnolo idiodianusm�-

twn, ¸ste na kataskeu�sete mÐa b�sh tou R3, kai breÐte ton pÐnaka thc apei-

kìnishc x 7→ Aix wc proc aut n th b�sh.

'Askhsh 1.21 JewroÔme dianusmatikì q¸ro V p�nw apì to s¸ma twn migadik¸n

arijm¸n, me peperasmènh di�stash, grammik apeikìnish L : V → V , kai grammikì

upìqwro X tou V , analloÐwto apì thn L. DeÐxte oti o X perièqei èna idiodi�nusma

thc L.

'Askhsh 1.22 'Estw L kai M dÔo grammikoÐ telestèc ston V , oi opoÐoi antime-

tatÐjentai: L◦M = M ◦L. DeÐxte oti tìte k�je idioq¸roc touM eÐnai analloÐwtoc

apì ton L.

Sumper�nete oti oi L kai M èqoun èna koinì idiodi�nusma.

'Askhsh 1.23 BreÐte tic idiotimèc enìc �nw trigwnikoÔ n× n pÐnaka.

'Askhsh 1.24 JewroÔme ton 3× 3 �nw trigwnikì pÐnaka

A =

λ1 1 0

0 λ2 1

0 0 λ3

.BreÐte ta idiodianusmatik� tou telest TA : K3 → K3 stic parak�tw peript¸seic:

1. λ1, λ2, λ3 eÐnai ana dÔo diaforetikoÐ.

2. λ1 6= λ2, λ2 = λ3.

3. λ1 = λ2 = λ3.


'Askhsh 1.25 JewroÔme ton 2×2 pÐnaka A =

[0 −1

1 0

]. 'Eqei o telest c TA :

R2 → R2 idiotimèc? SugkrÐnete autìn ton telest me ton antÐstoiqo TA : C2 → C2.

'Askhsh 1.26 DeÐxte oti, an o tetragwnikìc pÐnakac A me pragmatikoÔc sunte-

lestèc, ikanopoieÐ th sqèsh A2 + I = 0 autìc den epidèqetai pragmatikèc idiotimèc.

Sumper�nete oti den up�rqei 3× 3 pÐnakac me pragmatikoÔc suntelestèc o opoÐoc na

ikanopoieÐ th sqèsh A2 + I = 0.

'Askhsh 1.27 JewroÔme ton pÐnaka

A =

0 0 1

1 0 −1

−1 1 a

1. DeÐxte oti A3 − aA2 + 2A− I = 0.

2. DeÐxte oti o A eÐnai antistrèyimoc kai sumper�nete apì to aþ ton antÐstrofo

pÐnaka A−1.

3. UpologÐste ton pÐnaka A5 − aA4 +A3 − (1− a)A2 −A+ I.

'Askhsh 1.28 JewroÔme ton grammikì telest L : R3 → R3, tou opoÐou o

pÐnakac wc proc thn kanonik b�sh tou R3 eÐnai

A =

0 0 4

1 2 1

2 4 −2

.1. Diagwniopoi ste ton telest L.

2. BreÐte ton pÐnaka wc proc thn kanonik b�sh, tou telest L6−8L4+L3−9L+I.

'Askhsh 1.29 DÐdetai o pÐnakac A =

[6 −2

−2 9

]kai oi akoloujÐec un kai vn

oi opoÐec orÐzontai anadromik�:[u1

v1

]=

[1

1

]kai

[un+1

vn+1

]= A

[un

vn

].

1. Exet�ste e�n eÐnai o A diagwniopoi simoc

2. UpologÐste ta un kai vn wc sunart seic tou n.

3. BreÐte akoloujÐa wn tètoia ¸ste w1 = 1, w2 = 4 kai wn+2−15wn+1+50wn =

0.


'Askhsh 1.30 UpologÐste tic idiotimèc kai ta idiodianÔsmata tou pÐnaka

A =

2 2 −√

2

2 4 2√

2

−√

2 2√

2 1

kai diagwniopoi ste ton A.

'Askhsh 1.31 'Enac n× n pÐnakac A onom�zetai mhdenodÔnamoc e�n up�rqei

k�poioc fusikìc arijmìc k tètoioc ¸ste Ak = 0. DeÐxte oti e�n λ ∈ C eÐnai idiotim

enìc mhdenodÔnamou pÐnaka, tìte λ = 0. Sumper�nete oti k ≤ n.


Ask seic 2

Kef�laio 2

Nèoi DianusmatikoÐ Q¸roi

To kÔrio qarakthristikì thc Grammik c 'Algebrac eÐnai h dunatìthta na enopoieÐ se mÐa jew-

rÐa poll� diaforetik� majhmatik� antikeÐmena. 'Hdh èqoume dei paradeÐgmata dianusmatik¸n

q¸rwn sth GewmetrÐa, sth jewrÐa twn poluwnÔmwn, twn pin�kwn, twn sunart sewn prag-

matik¸n migadik¸n arijm¸n, kai �lla. Se autì to Kef�laio ja dieurÔnoume perissìtero

to pedÐo thc Grammik c 'Algebrac kai ja exet�soume dianusmatikoÔc q¸rouc p�nw apì pio

genik� sÔnola arijm¸n. EpÐshc ja melet soume diadikasÐec me tic opoÐec kataskeu�zoume

nèouc dianusmatikoÔc q¸rouc.

50

Kef�laio 2 Nèoi DianusmatikoÐ q¸roi 51

Di�lexh 9

Algebrik� s¸mata

Poièc idiìthtec twn pragmatik¸n kai twn migadik¸n arijm¸n qrhsimopoi same gia na a-

naptÔxoume th jewrÐa twn dianusmatik¸n q¸rwn mèqri t¸ra? Qrei�zetai na mporoÔme na

prosjètoume kai na afairoÔme touc suntelestèc se èna grammikì sunduasmì dianusm�twn,

na touc pollaplasi�zoume, kai na mporoÔme na touc diairoÔme me èna mh mhdenikì arijmì1.

Up�rqoun �lla sÔnola arijm¸n, ektìc apì touspragmatikoÔc kai touc migadikoÔc arijmoÔc,

pou èqoun autèc tic idiìthtec?

Sto sÔnolo twn rht¸n arijm¸n, Q, orÐzontai oi pr�xeic kai èqoun ìlec tic idiìthtec

pou qrhsimopoi same. MporoÔme, gia par�deigma, na efarmìsoume th diadikasÐa apaloif c

Gauss se ènan pÐnaka me rhtoÔc arijmoÔc, kai to apotèlesma ja eÐnai pÐnakac me rhtoÔc

arijmoÔc.

Autì den isqÔei sto sÔnolo twn akeraÐwn arijm¸n, Z. Den mporoÔme na diairèsoume me

opoiond pote mh mhdenikì akèraio kai na parameÐnoume sto sÔnolo twn akeraÐwn.

'Ena sÔnolo me dÔo pr�xeic pou èqoun tic basikèc idiìthtec twn pr�xewn thc prìsjeshc

kai tou pollaplasiasmoÔ pou qreiazìmaste sth melèth dianusmatik¸n q¸rwn, eÐnai èna

algebrikì s¸ma

Orismìc 2.1. 'Ena algebrikì s¸ma eÐnai èna sÔnolo K sto opoÐo orÐzontai dÔo

dimeleÐc pr�xeic, tic opoÐec onom�zoume prìsjesh kai pollaplasiasmì,

(a, b) 7→ a+ b kai (a, b) 7→ ab

kai oi opoÐec ikanopoioÔn ta akìlouja axi¸mata.

1H mình perÐptwsh pou qrhsimopoi same �llec idiìthtec twn arijm¸n, tan sthn eÔresh twn idiotim¸n

enìc grammikoÔ telest , pou qrei�sthke na broÔme tic rÐzec enìc poluwnÔmou


AS1. H prosetairistik idiìthta gia thn prìsjesh kai ton pollaplasiasmì: gia k�je

a, b, c ∈ K, isqÔoun

(a+ b) + c = a+ (b+ c) , (ab)c = a(bc)

AS2. H antimetajetik idiìthta gia thn prìsjesh kai ton pollaplasiasmì: gia k�je a, b ∈K, isqÔoun

a+ b = b+ a , ab = ba

AS3. H epimeristik idiìthta thc prìsjeshc wc proc ton pollaplasiasmì: gia k�je

a, b, c ∈ K, isqÔei

a(b+ c) = ab+ ac

AS4. Up�rqoun stoiqeÐa 0 ∈ K kai 1 ∈ K, tètoia ¸ste gia k�je a ∈ K,

a+ 0 = a kai a1 = a

AS5. Gia k�je a ∈ K up�rqei monadikì b ∈ K tètoio ¸ste a+ b = 0. To monadikì stoiqeÐo

b me aut thn idiìthta sumbolÐzetai −a kai onom�zetai antÐjeto tou a.

AS6. Gia k�je a ∈ K, a 6= 0, up�rqei monadikì b ∈ K tètoio ¸ste ab = 1. To monadikì

stoiqeÐo b me aut thn idiìthta sumbolÐzetai a−1 kai onom�zetai antÐstrofo tou a.

Par�deigma 2.1 Oi rhtoÐ arijmoÐ, Q, oi pragmatikoÐ arijmoÐ, R, kai oi migadikoÐ arijmoÐ,C, me tic gnwstèc pr�xeic thc prìsjeshc kai tou pollaplasiasmoÔ, apoteloÔn algebrik�

s¸mata. Oi akèraioi arijmoÐ, Z, den apoteloÔn s¸ma kaj¸c den ikanopoieÐtai to axÐwma

(AS6).

Par�deigma 2.2 To sÔnolo Z3 twn kl�sewn upoloÐpwn modulo 3 me thn prìsjesh kai

ton pollaplasiasmì modulo 3 (dec Jemèlia twn Majhmatik¸n, Kef�laio 2), apoteleÐ èna

s¸ma. Ta stoiqeÐa tou Z3 eÐnai oi kl�seic isodunamÐac thc sqèshc isotimÐac modulo 3 sto

Z:m ≡3 n e�n kai mìnon e�n m− n eÐnai pollapl�sio tou 3 .

ja sumbolÐsoume n3 thn kl�sh upoloÐpwn tou n. Aut h sqèsh diamerÐzei to sÔnolo twn

akeraÐwn se 3 kl�seic, 03, 13, 23. OrÐzoume pr�xeic prìsjeshc kai pollaplasiasmoÔ sto

sÔnolo Z3 wc ex c:

m3 + n3 = (m+ n)3 (2.1)

m3n3 = (mn)3 (2.2)

Gia par�deigma, 13 + 23 = 33 = 03, 2323 = 43 = 13.

Genikìtera, gia k�je pr¸to arijmì p, to sÔnolo Zp twn kl�sewn upoloÐpwn modulo p

apoteleÐ èna s¸ma.


'Askhsh 2.1 ApodeÐxte oti to sÔnolo Z6 twn kl�sewn upoloÐpwn modulo 6, den

eÐnai algebrikì s¸ma.

Upìdeixh: Exet�ste e�n to stoiqeÐo 36 èqei antÐstrofo.

Par�deigma 2.3 To sÔnolo twn rht¸n arijm¸n epektetamèno me thn tetragwnik rÐza

tou 2, dhlad to sÔnolo Q(√

2) = {a + b√

2 | a, b ∈ Q}, apoteleÐ èna s¸ma. OrÐzoume tic

pr�xeic prìsjeshc kai pollaplasiasmoÔ sto sÔnolo Q(√

2) akrib¸c ìpwc stouc pragmati-

koÔc arijmoÔc. To sÔnolo Q(√

2) eÐnai kleistì wc proc autèc tic pr�xeic, kai ètsi èqoume

kal� orismènec pr�xeic sto Q(√

2):

(a+ b√

2) + (c+ d√

2) = (a+ c) + (b+ d)√

2 ∈ Q(√

2)

(a+ b√

2) + (c+ d√

2) = (ac+ 2bd) + (ad+ bc)√

2 ∈ Q(√

2) .

'Askhsh 2.2 BreÐte to antÐstrofo tou a+ b√

2 ìtan b 6= 0 kai deÐxte oti an kei

sto Q(√

2).

Qrhsimopoi¸ntac ta axi¸mata mporoÔme na apodeÐxoume �llec idiìthtec pou èqei k�je

algebrikì s¸ma, ìpwc

1. Gia k�je a ∈ K, 0a = a0 = 0.

2. Gia k�je a, b ∈ K isqÔei a(−b) = (−a)b = −(ab).

MÐa shmantik idiìthta pou aporrèei apì ta axi¸mata enìc s¸matoc, kai sunep¸c isqÔei

se k�je s¸ma, en¸ mporeÐ na mhn isqÔei se �llec algebrikèc domèc, eÐnai h idiìthta thc

diagraf c.

L mma 2.1 Se èna s¸ma K, isqÔoun ta akìlouja:

1. E�n a, b ∈ K kai ab = 0, tìte eÐte a = 0 eÐte b = 0.

2. E�n a, b, c ∈ K kai c 6= 0, tìte ac = bc sunep�getai a = b.

Apìdeixh. aþ. 'Estw ab = 0 kai a 6= 0. Tìte, apì ta axi¸mata (AS6) kai (AS1), èqoume

a−1(ab) = (aa−1)b = 1b = b. All� afoÔ ab = 0, èqoume a−1(ab) = a0 = 0. 'Ara b = 0.

bþ. E�n ac = bc tìte ac + (−b)c = 0, �ra (a + (−b))c = 0. AfoÔ c 6= 0, apì to aþ èqoume

a+ (−b) = 0, sunep¸c a = b.

�

Algebrik� s¸mata ja melet soume pio analutik� sto m�jhma ��'Algebra I��.

Axi¸mata DianusmatikoÔ Q¸rou

Ja orÐsoume èna dianusmatikì q¸ro p�nw apì èna algebrikì s¸ma, wc èna sÔnolo me dÔo

pr�xeic, pou ikanopoioÔn ta kat�llhla axi¸mata.


Orismìc 2.2. JewroÔme èna algebrikì s¸ma K kai èna sÔnolo V me dÔo pr�xeic, thn

prìsjesh dianusm�twn,

α : V × V −→ V α(v, w) = v u w

kai ton pollaplasiasmì dianÔsmatoc me arijmì apì to s¸ma K,

µ : K× V −→ V µ(a, v) = a · v .

To sÔnolo V me tic pr�xeic α kai µ, onom�zetai dianusmatikìc q¸roc p�nw apì

to K e�n ikanopoioÔntai ta akìlouja axi¸mata.

DQ1. Gia k�je v, w ∈ V, v u w = w u v.

DQ2. Gia k�je v, w, u ∈ V, (v u w)u u = v u (w u u).

DQ3. Up�rqei stoiqeÐo 0 ∈ V tètoio ¸ste, gia k�je v ∈ V, v u 0 = v.

DQ4. Gia k�je v ∈ V up�rqei w ∈ V tètoio ¸ste uu w = 0.

DQ5. Gia k�je a, b ∈ K kai v ∈ V, a · (b · v) = (ab) · v.

DQ6. Gia k�je v ∈ V isqÔei 1 · v = v.

DQ7. Gia k�je a ∈ K kai v, w ∈ V, a · (v u w) = a · v u a · w.

DQ8. Gia k�je a, b ∈ K kai v, ∈ V, (a+ b) · v = a · v u b · v.

Ta stoiqeÐa enìc dianusmatikoÔ q¸rou onom�zontai dianÔsmata.

Parat rhsh: Sth diatÔpwsh twn axiwm�twn qrhsimopoioÔme to sÔmbolo + gia thn

prìsjesh sto s¸ma twn arijm¸n, kai to sÔmbolo u gia thn prìsjesh dianusm�twn. Argì-

tera den ja k�noume aut th di�krish, kaj¸c ja eÐnai safèc apì ta sumfrazìmena se poi�

pr�xh anaferìmaste. EpÐshc, e�n den up�rqei kÐndunoc sÔgqushc, ja qrhsimopoioÔme to Ðdio

sÔmbolo 0 eÐte gia ton arijmì mhdèn sto s¸ma, eÐte gia to mhdenikì di�nusma.

Pr¸ta apotelèsmata apì ta axi¸mata.

To mhdenikì di�nusma enìc q¸rou eÐnai monadikì, ìpwc blèpoume e�n upojèsoume oti 0 eÐnai

èna stoiqeÐo me thn idiìthta (DQ3). Tìte 0 = 0u 0 = 0.

L mma 2.2 JewroÔme èna dianusmatikì q¸ro V p�nw apì to s¸ma K, me pr�xeic u kai ·.

1. To ginìmeno enìc arijmoÔ a ∈ K, kai enìc dianÔsmatoc v ∈ V , eÐnai to mhdenikì

di�nusma e�n kai mìnon e�n a = 0 v = 0.

Pio analutik�, gia k�je v ∈ V, 0 · v = 0, kai gia k�je a ∈ K, a · 0 = 0, kai

antÐstrofa, gia k�je a ∈ K kai gia k�je v ∈ V , e�n a · v = 0, tìte eÐte a = 0 v = 0.


2. To antÐjeto enìc dianÔsmatoc v ∈ V eÐnai monadikì, kai Ðso me (−1) · v.

Apìdeixh. Gia to 1, jewroÔme èna di�nusma v ∈ V , kai ton arijmì mhdèn, 0 ∈ K. Ja

deÐxoume oti 0 · v = 0. 'Eqoume

0 · v = (0 + 0) · v

= 0 · v u 0 · v

'Estw w èna antÐjeto tou dianÔsmatoc 0 · v, dhlad 0 · v u w = 0. Tìte

0 = 0 · v u w

= (0 · v u 0 · v)u w

= 0 · v u (0 · v u w)

= 0 · v u 0

= 0 · v

JewroÔme ènan arijmì a ∈ K, kai to mhdenikì di�nusma 0 ∈ V . Ja deÐxoume oti a · 0 = 0.

'Eqoume

a · 0 = a · (0u 0)

= a · 0u a · 0

'Estw u èna antÐjeto tou dianÔsmatoc a · 0, dhlad a · 0 + u = 0. Tìte

0 = a · 0u u

= (a · 0u a · 0)u u

= a · 0u (a · 0u u)

= a · 0u 0

= a · 0

AntÐstrofa, e�n a 6= 0 kai a · v = 0, tìte

v = 1 · v = (a−1a) · v = a−1 · (a · v) = a−1 · 0 = 0 .

Gia to 2, upojètoume oti w kai w′ eÐnai antÐjeta tou v ∈ V , kai ja deÐxoume oti w = w′.

'Eqoume oti v u w = 0 = v u w′. Sunep¸c

w = w u 0

= w u (v u w′)

= (w u v)u w′

= (v u w)u w′

= 0u w′

= w′ u 0

= w′


T¸ra deÐqnoume oti to ginìmeno (−1) · v eÐnai antÐjeto tou v:

v u ((−1) · v) = (1 · v)u ((−1) · v)

= (1 + (−1)) · v

= 0 · v

= 0

To monadikì antÐjeto tou v sumbolÐzoume −v.�

ParadeÐgmata dianusmatik¸n q¸rwn p�nw apì èna s¸ma

K.

Sthn ��Eisagwg sth Grammik 'Algebra�� èqoume dei poll� paradeÐgmata dianusmatik¸n

q¸rwn p�nw apì touc pragmatikoÔc touc migadikoÔc arijmoÔc, ìpwc

1. Touc dianusmatikoÔc q¸rouc Rn kai Cn, twn diatetagmènwn n-�dwn pragmatik¸n

migadik¸n arijm¸n.

2. Touc dianusmatikoÔc q¸rouc RN kai CN, twn akolouji¸n me ìrouc pragmatikoÔc

migadikoÔc arijmoÔc, kai touc dianusmatikoÔc upoq¸rouc aut¸n, ìpwc to q¸ro twn

fragmènwn akolouji¸n, twn sugklinous¸n akolouji¸n.

3. Touc dianusmatikoÔc q¸rouc R[x] kai C[x], twn poluwnÔmwn mÐac metablht c me prag-

matikoÔc migadikoÔc suntelestèc, ìpwc kai touc upoq¸rouc Rk[x] kai Ck[x] twn

poluwnÔmwn bajmoÔ Ðsou mikrìterou apì k.

4. Touc dianusmatikoÔc q¸rouc RX kai CX , twn sunart sewn apì èna sÔnolo X stouc

pragmatikoÔc touc migadikoÔc arijmoÔc.

5. Touc dianusmatikoÔc q¸rouc M(m, n, R) kai M(m, n, C), twn m × n pin�kwn me

stoiqeÐa stouc pragmatikoÔc touc migadikoÔc arijmoÔc.

Aut� ta paradeÐgmata eÔkola genikeÔontai se dianusmatikoÔc q¸rouc p�nw apì �llo

algebrikì s¸ma.

Par�deigma 2.4 O dianusmatikìc q¸roc Zn3 , twn diatetagmènwn n-�dwn stoiqeÐwn tou

Z3 eÐnai dianusmatikìc q¸roc p�nw apì to s¸ma Z3. Gia par�deigma, o q¸roc Z23 apoteleÐtai

apì ta 9 diatetagmèna zeÔgh (m3, k3).

Ta shmeÐa (m3, k3) tou Z23 pou ikanopoioÔn thn exÐswshm3+k3 = 03 eÐnai ta {(03, 03), (13, 23), (23, 13)}.

Aut� ta trÐa shmeÐa apoteloÔn ènan upìqwro tou Z23, mÐa ��eujeÐa�� sto ��epÐpedo�� Z2

3.

To sÔnolo {(13, 23), (23, 13)} eÐnai grammik� exarthmèno, afoÔ (23, 13) = 23(13, 23).

To sÔnolo {(03, 23), (13, 13)} eÐnai grammik� anex�rthto, afoÔ mporoÔme na elègxoume

oti kanèna apì ta pollapl�sia tou (13, 13) me stoiqeÐa tou Z3 den eÐnai Ðso me to (03, 23).


Par�deigma 2.5 MporoÔme na jewr soume to sÔnolo twn pragmatik¸n arijm¸n wc

dianusmatikì q¸ro p�nw apì to s¸ma twn rht¸n arijm¸n Q, me tic sun jeic pr�xeic thc

prìsjeshc dÔo pragmatik¸n arijm¸n kai tou pollaplasiasmoÔ enìc pragmatikoÔ arijmoÔ

me èna rhtì. Ja qrhsimopoi soume ta gr�mmata x, y gia ta stoqeÐa tou R, pou ta jewroÔme

ta dianÔsmata, kai ta gr�mmata p, q gia ta stoiqeÐa tou Q ìtan ta jewroÔme touc arijmoÔc.

Gia k�je x ∈ R, tìte to sÔnolo ìlwn twn pollaplasÐwn tou x me èna rhtì arijmì,

apoteleÐ ènan upìqwro tou R, U = {px : p ∈ Q}. E�n x 6= 0, tìte o upìqwroc U èqei

di�stash 1.

E�n x, y eÐnai pragmatikoÐ arijmoÐ kai x/y den eÐnai rhtìc, to sÔnolo V = {px + qy :

p, q ∈ Q} eÐnai ènac upìqwroc tou R, di�stashc 2. Ta x kai y eÐnai grammik� anex�rthta

p�nw apì to Q, afoÔ den up�rqoun p kai q, ìqi kai ta dÔo Ðsa me to mhdèn, tètoia ¸ste

px+ qy = 0.


Di�lexh 10

EujÔ Ajroisma

Sthn Eisagwg sth Grammik 'Algebra eÐdame oti o grammikìc upìqwroc tou dianusmatikoÔ

q¸rou V pou par�getai apì thn ènwsh dÔo grammik¸n upoq¸rwn X kai Y , eÐnai to sÔnolo

ìlwn twn ajroism�twn enìc dianÔsmatoc sto X kai enìc dianÔsmatoc sto Y . Autì to

grammikì upìqwro onom�same �jroisma twn X kai Y , kai ton sumbolÐsame

X + Y = {x+ y : x ∈ X, y ∈ Y } .

MÐa perÐptwsh ajroÐsmatoc dÔo upoq¸rwn pou parousi�zei idiaÐtero endiafèron eÐnai ìtan

h tom twn X kai Y eÐnai tetrimmènh. E�n X ∩ Y = {0}, tìte k�je di�nusma u ∈ X + Y

gr�fetai me monadikì trìpo wc �jroisma dianusm�twn tou X kai tou Y : e�n x, x′ ∈ X,

y, y′ ∈ Y kai x + y = x′ + y′, tìte x = x′ kai y = y′. Se aut thn perÐptwsh onom�zoume

to dianusmatikì q¸ro X + Y (eswterikì) eujÔ �jroisma twn X kai Y , kai to sumbolÐzoume

X ⊕ Y .Se aut thn par�grafo ja melet soume mÐa pio genik kataskeu eujèwc ajroÐsmatoc.

Xekin�me me dÔo dianusmatikoÔc q¸rouc V kaiW , p�nw apì to Ðdio s¸maK, kai orÐzoume dom

dianusmatikoÔ q¸rou sto kartesianì ginìmeno V ×W . Me autì ton trìpo kataskeu�zoume

èna nèo dianusmatikì q¸ro, pou perièqei grammikoÔc upìqwrouc V ′ kai W ′, isomorfikoÔc me

touc V kai W , kai me thn idiìthta V ′ ∩W ′ = {0}.Autì to nèo dianusmatikì q¸ro ton onom�zoume (exwterikì) eujÔ �jroisma twn V kaiW ,

kai ton sumbolÐzoume V ⊕W . H qr sh tou Ðdiou onìmatoc kai tou Ðdiou sumbolismoÔ stic

dÔo diaforetikèc peript¸seic eÐnai dikaiologhmènh giatÐ, ìpwc ja apodeÐxome, to exwterikì

eujÔ �jroisma twn dianusmatik¸n q¸rwn V kai W eÐnai isomorfikì me to eswterikì eujÔ

�jroisma twn upoq¸rwn V ′ kai W ′,

V ⊕W ∼= V ′ +W ′ .

Orismìc 2.3. JewroÔme V kai W dianusmatikoÔc q¸rouc p�nw apì to s¸ma K. Sto

kartesianì ginìmeno V ×W = {(v, w) : v ∈ V, w ∈W} orÐzoume tic pr�xeic thc prìsjeshckai tou pollaplasiasmoÔ me stoiqeÐa tou K wc ex c: gia (v, w), (x, y) ∈ V ×W kai a ∈ K,

(v, w) + (x, y) = (v + x, w + y) kai a (v, w) = (a v, aw) .

Me autèc tic pr�xeic to sÔnolo V ×W eÐnai dianusmatikìc q¸roc p�nw apì to s¸ma K,

ton opoÐo onom�zoume (exwterikì) eujÔ �jroisma twn V kai W , kai sumbolÐzoume

V ⊕W .

Par�deigma 2.6 To eujÔ �jroisma R ⊕ R eÐnai o dianusmatikìc q¸roc pou sun jwc

sumbolÐzoume R2. StoiqeÐa tou eÐnai ta diatetagmèna zeÔgh pragmatik¸n arijm¸n (x, y), kai

oi pr�xeic orÐzontai kat� sunist¸sa.


Par�deigma 2.7 E�n V kai W eÐnai dÔo diaforetikoÐ dianusmatikoÐ q¸roi p�nw apì to

s¸ma K, to eujÔ �jroisma V ⊕W eÐnai diaforetikì apì to eujÔ �jroisma W ⊕ V . 'Omwc

ta dÔo ajroÐsmata eÐnai isomorfik�:

V ⊕W ∼= W ⊕ V .

Drasthriìthta 2.1 DeÐxte oti h apeikìnish (v, w) 7→ (w, v) eÐnai amfimonos -

manth kai grammik , kai sunep¸c orÐzei ènan isomorfismì C : V ⊕W −→W ⊕ V .

Par�deigma 2.8 E�n U, V kai W eÐnai dianusmatikoÐ q¸roi p�nw apì to s¸ma K, to

eujÔ �jroisma (U ⊕ V )⊕W kai to eujÔ �jroisma U ⊕ (V ⊕W ) eÐnai isomorfik�:

(U ⊕ V )⊕W ∼= U ⊕ (V ⊕W ) .

Drasthriìthta 2.2 DeÐxte oti h apeikìnish ((u, v), w)) 7→ (u, (v, w)) eÐnai

amfimonos manth kai grammik , kai sunep¸c orÐzei ènan isomorfismì

D : (U ⊕ V )⊕W −→ U ⊕ (V ⊕W ) .

Aut h parat rhsh mac epitrèpei na orÐsoume to eujÔ �jroisma perissìterwn apì dÔo

dianusmatik¸n q¸rwn, V1, V2, . . . , Vk, wc

k⊕j=1

Vj = (· · · ((V1 ⊕ V2)⊕ V3)⊕ · · ·)⊕ Vk ,

gnwrÐzontac oti e�n all�xoume tic parenjèseic ja èqoume ènan isìmorfo dianusmatikì q¸ro.

Par�deigma 2.9 Suqn� exet�zoume ènanm×n pÐnaka wc èna sÔnolo apì n st lec, k�je

mÐa apì tic opoÐec eÐnai èna di�nusma sto Kn. Aut h prosèggish mac odhgeÐ na jewr soume

to dianusmatikì q¸ro twn m× n pin�kwn, Mm,n(K) wc isomorfikì me to eujÔ �jroisma n

dianusmatik¸n q¸rwn Km,

Mm,n(K) ∼=n⊕j=1

Km .

To akìloujo L mma exhgeÐ th sqèsh metaxÔ tou eswterikoÔ kai tou exwterikoÔ eujèwc

ajroÐsmatoc.

L mma 2.3 E�n X kai Y eÐnai grammikoÐ upìqwroi tou V , kai X ∩ Y = {0}, tìte to

(eswterikì eujÔ) �jroisma twn X kai Y eÐnai isomorfikì me to (exwterikì) eujÔ �jroisma:

X + Y ∼= X ⊕ Y .


Apìdeixh. E�n v ∈ X + Y ⊆ V , up�rqoun monadik� x ∈ X kai y ∈ Y tètoia ¸ste

v = x + y. OrÐzoume thn apeikìnish L : X + Y −→ X ⊕ Y me L(v) = (x, y). Apì th

monadikìthta, h L eÐnai kal� orismènh. Elègqoume oti eÐnai amfimonos manth kai grammik .

�

Prosèxte th diafor� metaxÔ tou isomorfismoÔ sto L mma 2.3 kai tou isomorfismoÔ pou

prokÔptei apì thn epilog mÐac b�shc tou dianusmatikoÔ q¸rou V , V ∼= KdimV , (Eisagwg

sth Grammik 'Algebra, Je¸rhma ??). O isomorfismìc X + Y ∼= X ⊕ Y den basÐzetai se

k�poia epilog : ta x kai y eÐnai monadik� kajorismèna apì ta dedomèna tou probl matoc.

Lème oti autìc eÐnai ènac kanonikìc isomorfismìc, en¸ o isomorfismìc V ∼= KdimV

den eÐnai kanonikìc, afoÔ exart�tai apì thn epilog mÐac b�shc tou V .

Sto eujÔ �jroisma V ⊕W jewroÔme touc upìqwrouc V ′ = V ×{0} kai W ′ = {0}×W .

Oi apeikonÐseic j1 : V −→ V ′, j1(v) = (v, 0), kai j2 : W −→ W ′, j2(w) = (0, w) eÐnai

isomorfismoÐ. K�je di�nusma sto V ⊕W gr�fetai wc �jroisma enìc dianÔsmatoc sto V ′

kai enìc dianÔsmatoc sto W ′: (v, w) = (v, 0) + (0, w). Oi upìqwroi V ′ kai W ′ èqoun

tetrimmènh tom : e�n (v, w) ∈ V ′ ∩W ′ tìte (v, w) = (0, 0). Sunep¸c V ⊕W = V ′ +W ′.

L mma 2.4 E�n {v1, v2, . . . , vk} kai {w1, w2, . . . , wm} eÐnai grammik� anex�rthta sÔno-la (par�gonta sÔnola, b�seic) stouc dianusmatikoÔc q¸rouc V kai W antÐstoiqa, tìte

{(v1, 0), (v2, 0), . . . , (vk, 0), (0, w1), (0, w2), . . . , (0, wm)}

eÐnai grammik� anex�rthto sÔnolo (antÐstoiqa, par�gon sÔnolo, b�sh) tou V ⊕W .

Apìdeixh. Upojètoume oti ta stoiqeÐa a1, . . . , ak, b1, . . . , bm tou K ikanopoioÔn th

sqèsh

a1(v1, 0) + · · ·+ ak(vk, 0) + b1(0, w1) + · · ·+ bm(0, wm) = (0, 0) .

Tìte isqÔei (a1v1+· · ·+akvk, b1w1+· · ·+bmwm) = (0, 0), kai sunep¸c a1v1+· · ·+akvk = 0

kai b1w1 + · · ·+ bmwm = 0.

An {v1, . . . , vk} eÐnai grammik� anex�rthto sÔnolo sto V , sumperaÐnoume oti a1 = a2 =

· · · = ak = 0. AntÐstoiqa, an {w1, . . . , wm} eÐnai grammik� anex�rthto sÔnolo sto W ,

b1 = b2 = · · · = bm = 0. DeÐxame oti

{(v1, 0), (v2, 0), . . . , (vk, 0), (0, w1), (0, w2), . . . , (0, wm)}

eÐnai grammik� anex�rthto sÔnolo sto V ⊕W .

T¸ra jewroÔme stoiqeÐo (v, w) ∈ V ⊕W . An {v1, . . . , vk} eÐnai par�gon sÔnolo tou V ,

up�rqoun a1, . . . , ak ∈ K tètoia ¸ste v = a1v1 + · · ·+ akvk. AntÐstoiqa, an {w1, . . . , wm}eÐnai par�gon sÔnolo touW , up�rqoun b1, . . . , bm ∈ K tètoia ¸ste w = b1w1+ · · ·+bmwm.SumperaÐnoume oti

(v, w) = a1(v1, 0) + · · ·+ ak(vk, 0) + b1(0, w1) + · · ·+ bm(0, wm) ,


kai sunep¸c

{(v1, 0), (v2, 0), . . . , (vk, 0), (0, w1), (0, w2), . . . , (0, wm)}

eÐnai par�gon sÔnolo tou V ⊕W .

�

'Amesh sunèpeia tou L mmatoc eÐnai to akìloujo Je¸rhma:

Je¸rhma 2.5 E�n V kaiW eÐnai dianusmatikoÐ q¸roi peperasmènhc di�stashc p�nw apì

to s¸ma K, tìtedim(V ⊕W ) = dimV + dimW .

�

Par�deigma 2.10 E�n X kai Y eÐnai xèna sÔnola, X ∩ Y = ∅, tìte o dianusmatikìc

q¸roc twn apeikonÐsewn apì to sÔnolo X ∪ Y sto K, KX∪Y , eÐnai isomorfikìc me to eujÔ

�jroisma twn dianusmatik¸n q¸rwn twn apeikonÐsewn apì to X sto K kai apì to Y sto K,

KX∪Y ∼= KX ⊕KY .

Ja kataskeu�soume thn apeikìnish L : KX∪Y −→ KX ⊕ KY kai ja deÐxoume oti eÐnai

isomorfismìc dianusmatik¸n q¸rwn. JewroÔme f ∈ KX∪Y , dhlad apeikìnish f : X∪Y −→K. Tìte orÐzontai oi apeikonÐseic periorismoÔ thc f sta uposÔnola X kai Y , f |X : X −→ Kkai f |Y : Y −→ K. OrÐzoume L(f) = (f |X , f |Y ). H L eÐnai grammik . Gia par�deigma,

L(f + g) = ((f + g)|X , (f + g)|Y )

= (f |X + g|X , f |Y + g|Y )

= (f |X , f |Y ) + (g|X , g|Y ) .

Gia na deÐxoume oti h L eÐnai isomorfismìc, orÐzoume thn apeikìnish G : KX ⊕ KY −→KX∪Y , kai deÐqnoume oti eÐnai antÐstrofh thc L. Gia (f1, f2) ∈ KX ⊕ KY , orÐzoume

G(f1, f2) = f , ìpou f : X ∪ Y −→ K orÐzetai wc

f(t) =

{f1(t) e�n t ∈ X ,

f2(t) e�n t ∈ Y .

Profan¸c, f |X = f1 kai f |Y = f2, �ra L ◦ G(f1, f2) = (f1, f2). EpÐshc G ◦ L(f) =

G(f |X , f |Y ) = f . 'Ara G eÐnai antÐstrofh thc L, kai L eÐnai isomorfismìc.

Par�deigma 2.11 JewroÔme touc dianusmatikoÔc q¸rouc V1 kai V2 me b�seic B1 =

{v11, . . . v1k} kai B2 = {v21, . . . v2`} antÐstoiqa, kai touc dianusmatikoÔc q¸rouc W1 kai

W2 me b�seic C1 = {w11, . . . w1m} kai C2 = {w21, . . . w2n} antÐstoiqa.E�n L1 : V1 −→ W1 kai L2 : V2 −→ W2 eÐnai grammikèc apeikonÐseic, orÐzoume th

grammik apeikìnish

L1 ⊕ L2 : V1 ⊕ V2 −→W1 ⊕W2


h opoÐa apeikonÐzei to di�nusma (u1, u2) ∈ V1⊕V2 sto di�nusma (L1(u1), L2(u2)) ∈W1⊕W2.

E�n A eÐnai o pÐnakac thc L1 wc proc tic b�seic B1 kai C1, kai B eÐnai o pÐnakac thc L2

wc proc tic b�seic B2 kai C2, tìte o pÐnakac thc apeikìnishc L1 ⊕ L2 wc proc tic b�seic

{(v11, 0), . . . , (v1k, 0), (0, v21), . . . , (0, v2`)} tou V1 ⊕ V2 (2.3)

kai

{(w11, 0), . . . , (w1m, 0), (0, w21), . . . , (0, w2n)} tou W1 ⊕W2 (2.4)

eÐnai o pÐnakac [A 0

0 B

].

Me to eujÔ �jroisma dÔo dianusmatik¸n q¸rwn V kai W sundèontai oi akìloujec gram-

mikèc apeikonÐseic:

1. Oi kanonikèc emfuteÔseic tou V kai tou W sto V ⊕W ,

j1 : V −→ V ⊕W : v 7→ (v, 0)

j2 : W −→ V ⊕W : w 7→ (0, w) .

2. Oi kanonikèc probolèc tou V ⊕W epÐ twn V kai W ,

p1 : V ⊕W −→ V : (v, w) 7→ v

p2 : V ⊕W −→W : (v, w) 7→ w .

Par�deigma 2.12 JewroÔme touc q¸rouc V1, V2, W1 kai W2 tou ParadeÐgmatoc 2.11,

kai tic apeikonÐseic j1 : V1 −→ V1 ⊕ V2, j2 : V2 −→ V1 ⊕ V2 kai p1 : W1 ⊕W2 −→ W1 ,

p2 : W1 ⊕W2 −→W2.

E�n L : V1 ⊕ V2 −→ W1 ⊕W2 eÐnai grammik apeikìnish, tìte to pÐnakac thc L wc proc

tic b�seic 2.3 kai 2.4, eÐnai o [A C

D B

],

ìpou A eÐnai o pÐnakac thc apeikìnishc L11 = p1 ◦ L ◦ j1, C eÐnai o pÐnakac thc apeikìnishc

L12 = p1 ◦ L ◦ j2, D eÐnai o pÐnakac thc apeikìnishc L21 = p2 ◦ L ◦ j1 kai B eÐnai o pÐnakac

thc apeikìnishc L22 = p2 ◦ L ◦ j2.


Di�lexh 11

Q¸roc phlÐko

JewroÔme dianusmatikì q¸ro V p�nw apì to s¸ma K, kai grammikì upìqwro X tou V . Sto

V orÐzoume th sqèsh isodunamÐac

v ∼ w e�n kai mìnon e�n v − w ∈ X .

To sÔnolo twn kl�sewn isodunamÐac aut c thc sqèshc to onom�zoume phlÐko tou V me to

X, kai to sumbolÐzoume

V/X .

Thn kl�sh isodunamÐac tou v ∈ V wc proc aut th sqèsh th sumbolÐzoume

v +X, v .

Par�deigma 2.13 Sto R3, jewroÔme ton upìqwro X = {(t, t, 2t) | t ∈ R}. X eÐnai h

eujeÐa pou pern�ei apì ta shmeÐa (0, 0, 0) kai (1, 1, 2). H kl�sh isodunamÐac tou shmeÐou

(x, y, z) sto phlÐko R3/X eÐnai to sÔnolo twn dianusm�twn thc morf c

(x, y, z) + (t, t, 2t) t ∈ R,

dhlad eÐnai h eujeÐa pou pern�ei apì to (x, y, z) kai eÐnai par�llhlh proc ton X. To

sÔnolo phlÐko R3/X eÐnai to sÔnolo ìlwn twn eujei¸n sto R3 pou eÐnai Ðsec par�llhlec

me thn X.

Sto phlÐko V/X orÐzoume tic pr�xeic, gia v +X, y +X ∈ V/X, a ∈ K.

(v +X) + (w +X) = (v + w) +X

a (v +X) = a v +X .

L mma 2.6 Me autèc tic pr�xeic V/X eÐnai dianusmatikìc q¸roc p�nw apo to K.

O dianusmatikìc q¸roc V/X onom�zetai q¸roc phlÐko tou V mod X.

Apìdeixh. Mhdèn eÐnai h kl�sh tou X = 0+X kai to antÐjeto tou v+X eÐnai −(v+X) =

(−v) +X. EÔkola elègqoume ta upìloipa axi¸mata.

�

OrÐzetai kanonik epeikìnish P : V → V/X, me v 7→ v +X, h opoÐa eÐnai grammik : e�n

u, v ∈ V kai a ∈ K,

P (au+ v) = (au+ v) +X

= a(u+X) + (v +X)

= aP (u) + P (v) .


Je¸rhma 2.7 JewroÔme dianusmatikì q¸ro V peperasmènhc di�stashc kai upìqwro X

tou V . E�n {x1, . . . , xk} eÐnai b�sh tou X, kai {x1, . . . , xk, v1, . . . , vm} b�sh tou V , tìte

{v1 +X, . . . , vm +X} apoteleÐ b�sh tou V/X, kai sunep¸c

dim(V/X) = dimV − dimX .

Apìdeixh. Estw v ∈ V . Up�rqoun a1, . . . , ak kai b1, . . . , bm tètoia ¸ste v = a1x1 +

· · ·+ akxk + b1v1 + · · ·+ bmvm. Tìte v − (b1v1 + · · ·+ bmvm) ∈ X, �ra

v +X = (b1v1 + · · ·+ bmvm) +X

= b1(v1 +X) + · · ·+ bm(vm +X)

�ra {v1 +X, · · · , vm +X} par�goun to V/X.

'Estw b1(v1 +X) + · · ·+ bm(vm +X) = 0. Tìte b1v1 + · · ·+ bmvm ∈ X, �ra up�rqoun

a1, · · · , ak tètoia ¸ste b1v1+· · ·+bmvm = a1x1+· · ·+akxk. All� apo grammik anexarthsÐatwn {x1, · · · , xk, v1, · · · , vm} èqoume a1 = · · · = ak = b1 = · · · = bm = 0. 'Ara to sÔnolo

{v1 +X, · · · , vm +X} eÐnai grammik� anex�rthto kai apoteleÐ b�sh tou V/X.

�

Par�deigma 2.14 JewroÔme to �polÔedro� tou sq matoc, me mÐa èdra σ, pènte akmèc

α, β, γ, δ, ε kai tèssereic korufèc A, B, C, D.

OrÐzoume touc dianusmatikoÔc q¸rouc

C0 = {a1A+ a2B + a3C + a4D | ai ∈ R}

C1 = {b1α+ b2β + · · ·+ b5ε | bi ∈ R}

C2 = {sσ | s ∈ R} .

kai tic grammikèc apeikonÐseic

∂2 : C2 → C1, ∂1 : C1 → C0

me ∂2(σ) = α+ δ − ε kai

∂1(b1α+ · · ·+ b5ε) =

= b1(B −A) + b2(C −B) + b3(D − C) + b4(A−D) + b5(B −D)

= (b4 − b1)A+ (b1 − b2 + b5)B + (b2 − b3)C + (b3 − b4 − b5)D .

L mma 2.8 (L mma Poincare)

∂1∂2 = 0 .


Sq ma 2.1: 'Ena ��polÔedro��.

Apìdeixh. ∂1∂2(σ) = ∂1(α+ δ − ε) = (B −A) + (A−D) + (B −D) = 0.

�

Sunep¸c im ∂2 ⊆ ker ∂1 kai orÐzetai o dianusmatikìc q¸roc phlÐko

H1 = ker ∂1/ im ∂2 .

Ja prosdiorÐsoume mÐa b�sh touH1. Pr¸ta lÔnoume to sÔsthma twn exis¸sewn pou orÐzoun

to ker ∂1, kai brÐskoume oti ta dianÔsmata β + γ + ε kai α+ β + γ + δ apoteloÔn mÐa b�sh

tou q¸rou ker ∂1. To di�nusma α + δ − ε apoteleÐ mÐa b�sh tou im ∂2. Apì to Je¸rhma

2.7, gia na prosdiorÐsoume mÐa b�sh tou phlÐkou ker ∂1/ im ∂2, prèpei na broÔme mÐa b�sh

tou ker ∂1 h opoÐa na perièqei to di�nusma α+ δ − ε thc b�shc tou im ∂2. ParathroÔme oti

α + β + γ + δ = (α + δ − ε) + (β + γ + ε) kai sunep¸c {α + δ − ε, β + γ + ε} eÐnai b�shtou ker ∂1. SumperaÐnoume oti to di�nusma (β + γ + ε) + im ∂2 apoteleÐ b�sh tou H1.

H di�stash tou H1 metr�ei tic �trÔpec� sto polÔedro. To stoiqeÐo thc sugkekrimènhc

b�shc pou br kame diagr�fei ènan �kÔklo� gÔrw apì thn trÔpa tou poluèdrou.

Je¸rhma 2.9 (Je¸rhma IsomorfismoÔ) JewroÔme dianusmatikoÔc q¸rouc V kai

W p�nw apì to s¸ma K, kai grammik apeikìnish L : V −→W .

1. H apeikìnish L : V/ kerL −→W , L(v+kerL) = L(v), eÐnai kal� orismènh grammik

apeikìnish, kai h L paragontopoieÐtai wc sÔnjesh L = L ◦P , ìpou P : V −→ V/ kerL

eÐnai h kanonik epeikìnish v 7−→ v + kerL.

2. Up�rqei kanonikìc isomorfismìc

V/ kerL ∼= imL .

Apìdeixh. JewroÔme thn kl�sh isodunamÐac tou v sto V/ kerL, dhlad v+kerL = {u ∈V |u− v ∈ kerL}. ParathroÔme oti e�n u ∈ v + kerL tìte L(u) = L(v). Ara h apeikìnish


L : V/ kerL −→ W , L(v + kerL) = L(v) eÐnai kal� orismènh. Elègqoume oti h L eÐnai

grammik :

L (a(v + kerL) + (u+ kerL)) = L ((a v + u) + kerL)

= L(a v + u)

= aL(v) + L(u)

= a L(v + kerL) + L(u+ kerL) .

H L eÐnai monomorfismìc, ef� ìson e�n L(v) = L(u), tìte v − u ∈ kerL kai v + kerL =

u+kerL. H L eÐnai epeikonik sthn eikìna thc L, giatÐ e�n w = L(v), tìte w = L(v+kerL).

SumperaÐnoume oti L eÐnai isomorfismìc apì to phlÐko V/ kerL sthn eikìna imL.

�

Sth sunèqeia dÐdoume dÔo �lla apotelèsmata, ta opoÐa anafèrontai wc DeÔtero kai TrÐto

Je¸rhma IsomorfismoÔ.

Prìtash 2.10 (DeÔtero kai TrÐto Je¸rhma IsomorfismoÔ.)

1. JewroÔme dianusmatikì q¸ro V p�nw apì to s¸ma K, kaiX, Y grammikoÔc upìqwrouc

tou V . Tìte up�rqei kanonikìc isomorfismìc

(X + Y )/Y ∼= X/(X ∩ Y ) .

2. JewroÔme dianusmatikì q¸ro V p�nw apì to s¸ma K, kaiX, Y grammikoÔc upìqwrouc

tou V tètoiouc ¸steX ⊆ Y . Tìte Y/X eÐnai upìqwroc tou V/X, kai up�rqei kanonikìc

isomorfismìc

(V/X)/(Y/X) ∼= V/Y .

Apìdeixh. Gia thn apìdeixh tou DeÔterou Jewr matoc IsomorfismoÔ, jewroÔme to mo-

nomorfismì i : X −→ X + Y kai ton epimorfismì p : X + Y −→ (X + Y )/ Y . Ja deÐxoume

oti h sÔnjesh L = p ◦ i eÐnai epimorfismìc, me pur na X ∩ Y .Sugkekrimèna ja deÐxoume oti gia k�je (x+ y) + Y ∈ (X + Y )/ Y , L(x) = (x+ y) + Y .

Pr�gmati, afoÔ y ∈ Y , (x + y) + Y = x + Y = L(x). 'Ara L eÐnai epimorfismìc. E�n

x ∈ kerL, tìte L(x) = x+ Y = 0 + Y , dhlad x ∈ Y . 'Ara o pur nac eÐnai akrib¸c X ∩ Y .Apì to Je¸rhma IsomorfismoÔ,

(X + Y )/ Y ∼= X/ (X ∩ Y ) .

Gia thn apìdeixh tou TrÐtou Jewr matoc IsomorfismoÔ, jewroÔme dianusmatikoÔc q¸rouc

X ⊆ Y ⊆ V . Tìte profan¸c Y/X eÐnai uposÔnolo tou V/X. Gia na deÐxoume oti eÐnai

grammikìc upìqwroc, jewroÔme grammikì sunduasmì a(y1 + X) + (y2 + X) me y1, y2 ∈ Y .Autìc eÐnai Ðsoc me (ay1 + y2) +X kai sunep¸c an kei sto Y/X.

Sth sunèqeia jewroÔme thn antistoÐqish v + X 7−→ v + Y , gia v ∈ V . Aut dÐdei mÐa

kal� orismènh apeikìnish L : V/X −→ V/Y , afoÔ e�n v1 − v2 ∈ X, tìte v1 − v2 ∈ Y ,


kai sunep¸c e�n v1 + X = v2 + X, tìte L(v1 + X) = L(v2 + X). Ja deÐxoume oti h L

eÐnai epimorfismìc, me pur na Y/X. JewroÔme v + Y ∈ V/Y . Tìte v + X ∈ V/X kai

L(v+X) = v+Y , sunep¸c L eÐnai epimorfismìc. H kl�sh v+X an kei ston pur na thc L

e�n kai mìnon e�n L(v +X) = 0 + Y , dhlad v ∈ Y kai v +X ∈ Y/X. 'Ara kerL = V/Y .

Apì to Je¸rhma IsomorfismoÔ,

V/Y ∼= (V/X)/ (Y/X) .

�

Par�deigma 2.15 Up�rqei epÐshc isomorfismìc V ∼= kerL⊕ imL, all� autìc den eÐnai

kanonikìc. E�n epilèxoume mÐa b�sh {w1, . . . , wm} tou imL, kai v1, . . . , vm tètoia ¸ste

L(vi) = wi, tìte ta v1, . . . , vm eÐnai grammik� anex�rthta, kai orÐzetai grammik eneikìnish,

M2 : imL→ V : wi 7→ vi. H apeikìnish

M : kerL⊕ imL→ V : (v, w) 7→ v +M2(w)

eÐnai isomorfismìc, all� exart�tai apì thn epilog twn vi.


Di�lexh 12

DuðkoÐ q¸roi

Eqoume deÐ oti to sÔnolo twn grammik¸n apeikonÐsewn apì èna dianusmatikì q¸ro V se èna

dianusmatikì q¸ro W eÐnai epÐshc dianusmatikìc q¸roc, o L(U, V ). Sthn perÐptwsh pou

W eÐnai o monodi�statoc q¸roc K, onom�zoume ton L(V, K) duðkì q¸ro tou V , kai ton

sumbolÐzoume V ′.

Par�deigma 2.16 Sto dianusmatikì q¸ro Kn orÐzontai oi sunart seic suntetagmènwn

ϕ1, ϕ2, . . . , ϕn. E�n x = (x1, . . . , xn), tìte ϕk(x) = xk.

E�n {e1, . . . , en} eÐnai h kanonik b�sh tou Kn, èqoume ϕi(ej) = δij . E�n ψ : Kn −→ KeÐnai opoiad pote grammik sun�rthsh, h ψ kajorÐzetai apì tic timèc thc sta stoiqeÐa thc

b�shc: e�n ψ(ei) = ai ∈ K, tìte gia k�je x = (x1, . . . , xn), èqoume

ψ(x) = ψ(x1e1 + · · ·+ xnen)

= x1ψ(e1) + · · ·+ xnψ(en)

= x1a1 + · · ·+ xnan ,

dhlad , gia k�je grammik sun�rthsh ψ : Kn −→ K, ψ(x) eÐnai grammikìc sunduasmìc twn

suntetagmènwn xi = ϕi(x) tou x,

ψ(x) = x1a1 + · · ·+ xnan

= ϕ1(x)a1 + · · ·+ ϕn(x)an ,

all� kaj¸c o pollaplasiasmìc sto K eÐnai metajetikìc,

ψ(x) = a1ϕ1(x) + · · ·+ anϕn(x)

= (a1ϕ1 + · · ·+ anϕn)(x) .

Kaj¸c autì isqÔei gia k�je x ∈ Kn, èqoume ψ = a1ϕ1 + · · · + anϕn, kai oi sunart seic

suntetagmènwn ϕi par�goun to duðkì q¸ro (Kn)′.

Sumbolismìc. Ektìc apì to sunhjismèno sumbolismì twn sunart sewn, ϕ(x) = a, gia

stoiqeÐa tou duðkoÔ q¸rou qrhsimopoieÐtai kai o sumbolismìc

〈x, ϕ〉 = a .

Par�deigma 2.17 Sto q¸ro K[x] twn poluwnÔmwn mÐac metablht c, h apeikìnish ϕ :

K[x] −→ K gia thn opoÐa ϕ(p(x)) = 〈p(x), ϕ〉 = p(0) eÐnai èna stoiqeÐo tou duðkoÔ q¸rou

(K[x])′. Genikìtera, e�n t1, . . . , tk kai a1, . . . , ak eÐnai stoiqeÐa tou K, tìte h sun�rthsh

ψ : K[x] −→ K, gia thn opoÐa ψ(p(x)) = 〈p(x), ψ〉 = a1p(t1) + · · ·+ akp(tk), eÐnai stoiqeÐo

tou duðkoÔ q¸rou (K[x])′.


Par�deigma 2.18 Sto q¸ro C0[0, 1] twn suneq¸n sunart sewn sto kleistì di�sthma

[0, 1], e�n 0 ≤ a < b ≤ 1, kai α : [a, b] −→ R eÐnai suneq c, h sun�rthsh ψ : C0[0, 1] −→R, gia thn opoÐa

ψ(f) = 〈f, ψ〉 =

∫ b

aα(t) f(t) dt

eÐnai stoiqeÐo tou duðkoÔ q¸rou (C0[0, 1])′.

Par�deigma 2.19 JewroÔme mÐa diaforÐsismh sun�rthsh f : Rn −→ R. To diaforikì

thc f sto shmeÐo (x1, . . . , xn) eÐnai h grammik apeikìnish

Df(x1, . . . , xn) : Rn −→ R : (v1, . . . , vn) 7−→ v1∂f

∂x1+ · · ·+ vn

∂f

∂xn.

Dhlad Df(x1, . . . , xn) ∈ (Rn)′ kai Df eÐnai mÐa apeikìnish Df : Rn −→ (Rn)′, en gènei

mh grammik .

UpenjumÐzoume oti e�n V eÐnai dianusmatikìc q¸roc peperasmènhc di�stashc p�nw apì to

s¸ma K, {v1, . . . , vn} eÐnai b�sh tou V , kai a1, . . . , an ∈ K, tìte up�rqei monadik grammik

apeikìnish ϕ ∈ L(V, K) tètoia ¸ste ϕ(vi) = ai gia k�je i = 1, . . . , n. Dhlad up�rqei

monadikì stoiqeÐo ϕ ∈ V ′ tètoio ¸ste 〈vi, ϕ〉 = ai.

Je¸rhma 2.11 E�n V eÐnai dianusmatikìc q¸roc peperasmènhc di�stashc kai

{v1, . . . , vn} eÐnai b�sh tou V , tìte up�rqei b�sh tou V ′, {ϕ1, . . . , ϕn}, gia thn opoÐa,

me to sumbolismì δi j tou Kronecker,

ϕj(vi) = 〈vi, ϕj〉 = δij

kai �ra

dimV ′ = dimV .

H b�sh {ϕ1, . . . , ϕn} onom�zetai duðk b�sh thc {v1, . . . , vn}.Apìdeixh. Pr¸ta deÐqnoume oti to {ϕ1, . . . , ϕn} eÐnai grammik� anex�rthto. O grammikìc

sunduasmìc ψ = a1ϕ1 + · · · + anϕn eÐnai 0 e�n kai mìnon e�n ψ(v) = 0 gia k�je v ∈ V .Eidikìtera, gia k�je i = 1, . . . , n, èqoume

0 = ψ(vi) = 〈vi, a1ϕ1 + · · ·+ anϕn〉

= a1〈vi, ϕ1〉+ · · ·+ an〈vi, ϕn〉

= a1δi1 + · · ·+ anδin

= ai

kai sumperaÐnoume oti {ϕ1, . . . , ϕn} eÐnai grammik� anex�rthto.

Gia na deÐxoume oti ta ϕ1, . . . , ϕn par�goun to duðkì q¸ro V ′, jewroÔme ψ ∈ V ′ me

〈vi, ψ〉 = ci gia k�je i = 1, . . . , n, kai u = b1v1 + · · ·+ bnvn. Tìte

〈u, ψ〉 = 〈b1v1 + · · ·+ bnvn, ψ〉


= b1〈v1, ψ〉+ · · ·+ bn〈vn, ψ〉

= b1c1 + · · ·+ bncn

= 〈u, ϕ1〉c1 + · · ·+ 〈u, ϕn〉cn= 〈u, c1ϕ1 + · · ·+ cnϕn〉

�ra ψ = c1ϕ1 + · · ·+ cnϕn.

�

Je¸rhma 2.12 E�n v, w ∈ V kai v 6= w, tìte up�rqei ψ ∈ V ′ tètoio ¸ste 〈v, ψ〉 6=〈w, ψ〉.

Apìdeixh. JewroÔme b�sh {vi, . . . , vn} tou V , kai th duðk b�sh {ϕ1, . . . , ϕn} tou

V ′. E�n 〈v, ψ〉 = 〈w, ψ〉 gia ìla ta ψ ∈ V ′, tìte, gia k�je ϕi thc duðk c b�shc, èqoume

〈v − w, ϕi〉 = 0, kai v − w = 〈u− w, ϕ1〉v1 + · · ·+ 〈v − w, ϕn〉vn = 0.

�

AfoÔ o duðkìc q¸roc V ′ eÐnai dianusmatikìc q¸roc p�nw apì to s¸ma K, mporoÔme na

jewr soume to duðkì tou q¸ro, (V ′)′, o opoÐoc sumbolÐzetai V ′′. 'Ena stoiqeÐo χ tou q¸rou

V ′′ eÐnai mÐa grammik sun�rthsh sto q¸ro V ′,

χ : V ′ −→ K : ψ 7−→ χ(ψ) .

Par�deigma 2.20 E�n v ∈ V , tìte h apeikìnish η : ψ 7−→ 〈v, ψ〉 eÐnai grammik wc procto ψ, (Askhsh: TÐ akrib¸c shmaÐnei autì?). Sunep¸c gia k�je v ∈ V orÐzetai, me fusikì

trìpo, èna η ∈ V ′′. Ja doÔme oti, gia q¸rouc peperasmènhc di�stashc, aut h antistoiqÐa

eÐnai ènac isomorfismìc.

Je¸rhma 2.13 E�n V eÐnai dianusmatikìc q¸roc peperasmènhc di�stashc, tìte h apei-

kìnish

ν : V −→ V ′′ : v 7−→ (η : ψ 7−→ 〈v, ψ〉)

eÐnai (kanonikìc) isomorfismìc.

Apìdeixh. H ν eÐnai grammik :

ν(a v + w)(ψ) = 〈a v + w, ψ〉

= a 〈v, ψ〉+ 〈w, ψ〉

= a ν(v)(ψ) + ν(w)(ψ) .

Gia na deÐxoume oti h ν eÐnai eneikìnish, jewroÔme v, w ∈ V . E�n ν(v) = ν(w) tìte, gia

k�je ψ ∈ V ′, ψ(v) = ψ(w), kai apì to Je¸rhma 2.12, v = w.

Apì to Je¸rhma 2.11, dimV ′′ = dimV ′ = dimV , kai sunep¸c h ν eÐnai epeikìnish.

�

E�n L : V −→ W eÐnai grammik apeikìnish, mporoÔme na orÐsoume th duðk apeikì-

nish L′ ( an�strofh apeikìnish LT ) an�mesa stouc duðkoÔc q¸rouc:

L′ : W ′ −→ V ′ , L′(ψ) = ψ ◦ L .


Prosèxte oti h L′ èqei for� antÐjeth apì thn L, kai ikanopoieÐ th sqèsh 〈v, L′ψ〉 = 〈Lv, ψ〉.ParathroÔme oti h antistoiqÐa L 7→ L′ eÐnai grammik apeikìnish apì to L(V, W ) sto

L(W ′, V ′): (aL+M)′ = aL′ +M ′.

L mma 2.14 JewroÔme tic grammikèc apeikonÐseic L : U −→ V kaiM : V −→W . Tìte

1. (M ◦ L)′ = L′ ◦M ′.

2. E�n h L : U −→ V eÐnai antistrèyimh, tìte h L′ : V ′ −→ U ′ eÐnai epÐshc antistrè-

yimh kai (L−1)′ = (L′)−1.

3. E�n oi q¸roi U kai V eÐnai peperasmènhc di�stashc, tìte to akìloujo di�gramma

grammik¸n apeikonÐsewn metatÐjetai:

U −→ V

νU ↓ ↓ νV

U ′′ −→ V ′′,

dhlad

νV ◦ L = L′′ ◦ νU ,

ìpou L′′ = (L′)′ kai νU , νV eÐnai oi kanonikoÐ isomorfismoÐ.

Apìdeixh. E�n ζ ∈W ′, tìte

(M ◦ L)′(ζ) = ζ ◦ (M ◦ L)

= (ζ ◦M) ◦ L

= L′(ζ ◦M)

= L′(M ′(ζ))

= L′ ◦M ′(ζ)

E�n h L eÐnai antistrèyimh, tìte(L−1

)′ ◦ L′ = (L ◦ L−1)′ = (IV )′ = IV ′

kai

L′ ◦(L−1

)′=(L−1 ◦ L

)′= (IU )′ = IU ′ .

E�n u ∈ U kai ϕ ∈ U ′, èqoume νU (u)(ϕ) = 〈u, ϕ〉. E�n ψ ∈ V ′, èqoume

L′′νU (u)(ψ) = νU (u)(L′ψ)

= 〈u, L′ψ〉

= 〈Lu, ψ〉

= νV (Lu)(ψ)

= νV ◦ L(u)(ψ) .


�

PÐnakac duðk c apeikìnishc

JewroÔme dianusmatikoÔc q¸rouc V kaiW , me b�seic B = {v1, . . . , vn} kai C = {w1, . . . , wm}antÐstoiqa.

GnwrÐzoume oti orÐzetai h duðk b�sh B′ = {ϕ1, . . . , ϕn} tou duðkoÔ q¸rou V ′, ìpou

ϕi(vj) = 〈vj , ϕi〉 = δi j gia i, j = 1, . . . , n

kai h duðk b�sh C′ = {ψ1, . . . , ψm} tou duðkoÔ q¸rou W ′, ìpou

ψk(w`) = 〈w`, ψk〉 = δk ` gia k, ` = 1, . . . , m .

JewroÔme grammik apeikìnish L : V −→W kai ton pÐnaka A = CLB thc apeikìnishc L

wc proc tic b�seic B tou V kai C tou W . H duðk apeikìnish L′ : W ′ −→ V ′ èqei pÐnaka

wc proc tic b�seic C′ kai B′, A′ = B′L′C′ = (a′j k) tètoio ¸ste

L′(ψk) =

n∑j=1

a′j kϕj . (2.5)

'Omwc gnwrÐzoume oti h duðk apeikìnish ikanopoieÐ, gia k�je k = 1, . . . , m kai k�je v =

b1v1 + · · ·+ bnvn,

L′(ψk)(v) = ψk ◦ L(v)

= ψk(Av)

= ψk

m∑`=1

n∑j=1

a` jbjw`

=

n∑j=1

m∑`=1

a` jbjψk(w`)

=

n∑j=1

ak jbj afoÔ ψk(w`) = 0 ìtan ` 6= k,

=

n∑j=1

ak jϕj(v)

SumperaÐnoume oti

L′(ψk) =

n∑j=1

ak jϕj . (2.6)


SugkrÐnontac tic 2.5 kai 2.6, katal goume oti a′j k = ak j , dhlad oti o pÐnakac A′ = B′L

′C′

thc duðk c apeikìnishc L′ : W ′ −→ V ′ eÐnai o an�strofoc tou pÐnaka A = CLB thc

apeikìnishc L : V −→W .

Par�deigma 2.21 JewroÔme pÐnaka A =

[1 2 −1

0 2 1

]kai thn antÐstoiqh grammik

apeikìnish TA : R3 −→ R2.

Oi sunart seic suntetagmènwn orÐzoun b�seic stouc duðkoÔc q¸rouc (R3)′ kai (R2)′:

e�n E3 = {e1, e2, e3} kai E2 = {f1, f2} eÐnai oi kanonikèc b�seic twn R3 kai R2, oi duðkèc

b�seic eÐnai E ′3 = {ϕ1, ϕ2, ϕ3} kai E ′2 = {ψ1, ψ2}, ìpou ϕi(ej) = δi j gia i, j = 1, 2, 3 kai

ψk(f`) = δk ` gia k, ` = 1, 2.

H duðk apeikìnish (TA)′ : (R2)′ −→ (R3)′ apeikonÐzei th sun�rthsh ψ ∈ (R2)′ me

〈fk, ψ〉 = bk gia k = 1, 2, sth sun�rthsh (TA)′(ψ) ∈ (R3)′ me 〈ei, (TA)′(ψ)〉 = ci gia

i = 1, 2, 3, kai c1

c2

c3

=

1 0

2 2

−1 1

[ b1

b2

].

Dhlad , e�n ψ(y1, y2) = b1y1 + b2y2, tìte

(TA)′(ψ)(x1, x2, x3) = c1x1 + c2x2 + c3x3 = b1x1 + 2(b1 + b2)x2 + (b2 − b1)x3 .


'Askhsh 2.3 Jewr ste ta dianÔsmata x, y, u kai v sto K4, ìpou K eÐnai èna

s¸ma sto opoÐo 1 6= −1, kai touc upìqwrouc Z kai W pou par�gontai apo ta

sÔnola {x, y} kai {u, v} antÐstoiqa. Se poièc apo tic akìloujec peript¸seic isqÔei

oti K4 = Z ⊕W .

aþ.

x = (1, 1, 0, 0) y = (1, 0, 1, 0)

u = (0, 1, 0, 1) v = (0, 0, 1, 1)

bþ.

x = (1, 0, 0, 1) y = (0, 1, 1, 0)

u = (1, 0, −1, 0) v = (0, 1, 0, 1)

'Askhsh 2.4 E�n X, Y, Z eÐnai dianusmatikoÐ q¸roi p�nw apo to s¸ma K, deÐxte

oti up�rqoun isomorfismoÐ.

aþ. X ⊕ Y ∼= Y ⊕X

bþ. X ⊕ (Y ⊕ Z) ∼= (X ⊕ Y )⊕ Z

'Askhsh 2.5 'Estw dianusmatikìc q¸roc U kai upìqwroi X1, X2, . . . , Xn tou

U . JewroÔme touc dianusmatikoÔc q¸rouc Y = X1 + X2 + · · · + Xn kai V =

X1 ⊕X2 ⊕ · · · ⊕Xn.

aþ. DeÐxte oti Y ∼= V e�n kai mìnon e�n gia k�je i = 1, 2, . . . , n − 1 isqÔei

(X1 + · · ·+Xi) ∩Xi+1 = 0.

bþ. DeÐxte oti e�n Y ∼= V tìte Xi ∩Xj = 0 gia k�je i 6= j. BreÐte èna par�deigma

me treÐc upìqwrouc X1, X2, X3 gia na deÐxete oti den isqÔei to antÐstrofo.

'Askhsh 2.6 E�n L : V → X kai M : W → Y eÐnai grammikèc apeikonÐseic

dianusmatik¸n q¸rwn p�nw apì to s¸ma K, deÐxte oti orÐzetai apeikìnish L ⊕Mapì to V ⊕W sto X ⊕ Y ,

L⊕M(v, w) = (L(v), M(w)) ,

h opoÐa eÐnai grammik . DeÐxte oti im (L⊕M) = imL⊕ imM kai oti ker(L⊕M) =

kerL⊕ kerM .

'Askhsh 2.7 Elègxte oti oi kanonikèc emfuteÔseic j1 : V −→ V ⊕W kai j2 :

W −→ V ⊕W kai oi kanonikèc probolèc p1 : V ⊕W −→ V kai p2 : V ⊕W −→W

eÐnai grammikèc apeikonÐseic, kai oti ikanopoioÔn tic sqèseic

p1 ◦ j1 = IV , p1 ◦ j2 = 0 , p2 ◦ j1 = 0 , p2 ◦ j2 = IW .


'Askhsh 2.8 Jewr ste to dianusmatikì q¸ro R[x] ìlwn twn poluwnÔmwn mÐac

metablht c, kai ton upìqwro Pn twn poluwnÔmwn bajmoÔ mikrìterou Ðsou me n.

'Eqei o q¸roc phlÐko R[x]/Pn peperasmènh di�stash?

'Askhsh 2.9 Sto dianusmatikì q¸ro C4 jewr ste touc upìqwrouc

U ={

(z1, z2, z3, z4) ∈ C4 : z1 = z2},

V ={(z1, z2, z3, z4) ∈ C4 : z1 − z2 − iz3 + iz4 = 2z2 + z3 = 2z2 + (1 + i)z3 − iz4 = 0

}.

aþ. DeÐxte oti V ⊆ U .

bþ. BreÐte mÐa b�sh tou q¸rou phlÐko U/V .

'Askhsh 2.10 Upojètoume oti L : V −→ W eÐnai grammik apeikìnish, X eÐnai

grammikìc upìqwroc tou V , Y eÐnai grammikìc upìqwroc tou W , kai isqÔei L(X) ⊆Y . DeÐxte oti orÐzetai grammik apeikìnish L : V/X −→W/Y , tètoia ¸ste

L ◦ P = Q ◦ L ,

ìpou P : V −→ V/X kai Q : W −→W/Y eÐnai oi kanonikèc epeikonÐseic.


'Askhsh 2.11 JewroÔme ton dianusmatikì q¸ro V = R[x] twn poluwnÔmwn mÐac

metablht c, me pragmatikoÔc suntelestèc, kai to sÔnoloW = {s(x) ∈ R[x] : s(x) =

q(x)(x2 + 1), q(x) ∈ R[x]} twn poluwnÔmwn pou diairoÔntai me to x+1.

aþ. DeÐxte oti W eÐnai grammikìc upìqwroc tou V .

bþ. DeÐxteo ti gia k�je polu¸numo p(x) ∈ V up�rqei polu¸numo r(x) bajmoÔ Ðsou

mikrìterou apì 1, tètoio ¸ste p(x)− r(x) ∈W .

gþ. Jewr ste to q¸ro phlÐko V/W . DeÐxte oti k�je polu¸numo p(x) ∈ V an kei

se mÐa kl�sh isodunamÐac (ax+ b) +W , gia a, b ∈ R.

dþ. DeÐxte oti B = {x+W, 1 +W} apoteleÐ b�sh tou q¸rou phlÐko V/W . (Sto

gþ deÐxate oti ta stoiqeÐa x + W , 1 + W par�goun to q¸ro V/W . Apomènei

na deÐxete oti eÐnai grammik� anex�rthta.)

Jewr ste ton pÐnaka A =

[0 −1

1 0

]. Gia k�je polu¸numo p(x) ∈ V , orÐzetai o

pÐnakac p(A) ∈ M2(R). Jewr ste to sÔnolo twn pin�kwn U = {p(A) ∈ M2(R) :

p(x) ∈ V }.

eþ. DeÐxte oti U eÐnai dianusmatikìc q¸roc.

�þ. DeÐxte oti h apeikìnish F : V/W −→ U : p(x) + W 7−→ p(A) eÐnai kal�

orismènh kai eÐnai (kanonikìc) isomorfismìc V/W ∼= U .

'Askhsh 2.12 BreÐte èna mh mhdenikì stoiqeÐo ϕ tou q¸rou (C3)′, tètoio ¸ste

e�n x1 = (1, 1, 1) kai x2 = (1, 1, −1), tìte 〈x1, ϕ〉 = 〈x2, ϕ〉 = 0.

'Askhsh 2.13 Ta dianÔsmata x1 = (1, 1, 1), x2 = (1, 1, −1) kai x3 =

(1, −1, −1) apoteloÔn b�sh tou C3. E�n {ϕ1, ϕ2, ϕ3} eÐnai h duðk b�sh tou (C3)′,

kai x = (0, 1, 0), breÐte ta 〈x, ϕ1〉, 〈x, ϕ2〉 kai 〈x, ϕ3〉.

'Askhsh 2.14 Poièc apì tic akìloujec sunart seic sto C3 eÐnai stoiqeÐa tou

(C3)′?

aþ. ϕ(z1, z2, z3) = z1 + 3z2

bþ. ϕ(z1, z2, z3) = z1 − z32

gþ. ϕ(z1, z2, z3) = z2 + 1

dþ. ϕ(z1, z2, z3) = z1 + z2z3


'Askhsh 2.15 JewroÔme mÐa akoloujÐa pragmatik¸n arijm¸n (ck) =

(c0, c1, c2, . . .). Gia k�je polu¸numo p(x) = a0 + a1x + a2x2 + · · · + anx

n ∈ R[x]

orÐzoume ψ(p) =∑n

i=0 aici. DeÐxte oti ψ ∈ (R[x])′, kai oti k�je stoiqeÐo tou duðkoÔ

q¸rou (R[x])′ prokÔptei me autìn ton trìpo, gia kat�llhlh epilog thc akoloujÐac

(ak).

'Askhsh 2.16 E�n ψ ∈ V ′, ψ 6= 0 kai a ∈ K, eÐnai al jeia oti up�rqei x ∈ Vtètoio ¸ste 〈x, ψ〉 = a?

'Askhsh 2.17 JewroÔme dianusmatikoÔc q¸rouc U , V kai W , kai grammik apei-

kìnish L : V → W . DeÐxte oti h apeikìnish L(U, V ) → L(U, W ) : M 7→ L ◦MeÐnai grammik .

'Askhsh 2.18 DeÐxte oti e�n V eÐnai dianusmatikìc q¸roc, kai ϕ eÐnai mh mhdenikì

stoiqeÐo touV ′, tìte to sÔnolo

U = {x ∈ V |〈x, ϕ〉 = 0}

eÐnai upìqwroc tou V . E�n dimV <∞, breÐte th di�stash tou U .

'Askhsh 2.19 DeÐxte oti e�n ϕ kai ψ ∈ V ′ kai gia k�je x ∈ V , 〈x, ϕ〉 = 0 e�n

kai mìnon e�n 〈x, ψ〉 = 0, tìte up�rqei a ∈ K tètoio ¸ste ψ = aϕ.

'Askhsh 2.20 MÐa sun�rthsh Q : W × V → R onom�zetai digrammik e�n

h Q eÐnai grammik wc proc k�je metablht qwrist�, dhlad , gia k�je w, z ∈ W ,

u, v ∈ V kai a ∈ K, isqÔoun ta akìlouja:

Q(w + z, u) = Q(w, u) +Q(z, u) Q(aw, u) = aQ(w, u)

Q(w, u+ v) = Q(w, u) +Q(w, v) Q(w, au) = aQ(w, u)

DeÐxte oti to sÔnolo L(W, V ; K) twn digrammik¸n sunart sewn sto W × V eÐnai

dianusmatikìc q¸roc wc proc tic kat� shmeÐo pr�xeic.

'Askhsh 2.21 JewroÔme mÐa apeikìnish L : W → V ′, dhlad gia k�je w ∈ W ,

L(w) eÐnai mÐa grammik apeikìnish L(w) : V → R.

aþ. Ti shmaÐnei na eÐnai h L grammik apeikìnish?

bþ. DeÐxte oti e�n L eÐnai grammik , tìte h apeikìnish M : W × V → R, gia thn

opoÐa M(w, v) = 〈v, L(w)〉 eÐnai digrammik .

gþ. DeÐxte oti h antistoiqÐa L 7→M orÐzei isomorfismì metaxÔ tou dianusmatikoÔ

q¸rou L(W, V ′) kai tou q¸rou twn digrammik¸n sunart sewn L(W,V ;R).


Di�lexh 13

PÐnakec p�nw apì to s¸ma K

To sÔnolo twn m × n pin�kwn me ìrouc sto s¸ma K sumbolÐzetaiM(m, n, K) Km,n

Mm,n(K). To sÔnolo twn tetragwnik¸n n× n pin�kwn me ìrouc sto s¸ma K to sumbolÐ-

zoumeM(n, K).

H jewrÐa twn pin�kwn pou melet same sthn Eisagwg sth Grammik 'Algebra, sto me-

galÔtero mèroc thc isqÔei epakrib¸c gia pÐnakec me ìrouc se opoiod pote s¸ma.

Gia opoiod pote s¸ma K,M(m, n, K) eÐnai dianusmatikìc q¸roc, kai o pollaplasiasmìc

pin�kwn orÐzetai ìtan autoÐ èqoun kat�llhlo sq ma. E�n A eÐnai m×n pÐnakac, kai B eÐnai

n× k pÐnakac p�nw apì to s¸ma K, orÐzetai to ginìmeno AB, kai eÐnai o m× k pÐnakac C,

o opoÐoc èqei sth jèsh i j, dhlad sthn i gramm kai sthn j st lh, ton ìro

ci j = ai 1 b1 j + · · ·+ ai n bn j

=n∑`=1

ai ` b` j .

H apaloif Gauss mporeÐ epÐshc na efarmosteÐ se opoiod pote s¸ma K gia na metatrèyoume

èna m× n pÐnaka se èna grammoðsodÔnamo pÐnaka se klimakwt morf .

H t�xh tou pÐnaka A ( bajmìc tou pÐnaka A) eÐnai o arijmìc r(A),

r(A) = arijmìc grammik� anex�rthtwn gramm¸n tou A

= arijmìc grammik� anex�rthtwn sthl¸n tou A

= arijmìc odhg¸n sto grammoðsodÔnamo pÐnaka se klimakwt morf

Prosèxte oti se èna pÐnaka se klimakwt morf , k�je m mhdenik gramm èqei ènan odh-

gì. (KamÐa for� mac diafeÔgei o odhgìc sthn teleutaÐa gramm , epeid den qrhsimopoieÐtai

kat� thn apaloif ).

Prìtash 2.15 K�je grammik apeikìnish L : Kn → Km antistoiqeÐ se èna m×n pÐnaka

A, tètoio ¸ste, gia k�je b ∈ Kn, L(b) = Ab,

L(b1, . . . , bn) =

a11 · · · a1n...

. . ....

am1 · · · amn

b1...

bn

Apìdeixh. JewroÔme thn kanonik b�sh {e1, . . . , en} tou Kn,

e1 = (1, 0, . . . , 0), . . . , ej = (δ1j , . . . , δnj), . . . , en = (0, . . . , 0, 1)

kai ta dianÔsmata L(e1), . . . , L(en) ∈ Km.


OrÐzoume ton pÐnaka A na èqei sth j st lh, gia j = 1, . . . , n, to di�nusma L(ej) ∈ Km.

Dhlad

A =

a11 · · · a1n...

. . ....

am1 · · · amn

ìpou (a1j , . . . , amj) = L(ej).

E�n b = (b1, . . . , bn) èqoume b = b1e1 + · · ·+ bnen, kai sunep¸c

L(b) = b1L(e1) + · · ·+ bnL(en)

= b1

a11...

am1

+ · · ·+ bn

a1n...

amn

.UpenjumÐzoume thn par�stash tou ginomènou pÐnaka me di�nusma wc grammikì sunduasmì

twn sthl¸n tou pÐnaka, kai èqoume

L(b) =

a11 · · · a1n...

. . ....

am1 · · · amn

b1...

bn

= Ab .

�

Q¸roi grammik¸n apeikonÐsewn

Na sumplhrwjeÐ?

******

EpÐshc, parat rhsh gia paragontopoi sh apeikìnishc. Na dw pou ja p�ei.

Sth Grammik 'Algebra I, èqoume deÐ oti e�n A eÐnai m × n pÐnakac, sth grammik apei-

kìnish TA : Rn → Rm : x 7→ Ax antistoiqeÐ mÐa antistrèyimh apeikìnish apì to q¸ro

gramm¸n R(AT ) sto q¸ro sthl¸n R(A), kai oti h TA mporeÐ na ekfrasteÐ wc sÔnjesh

tri¸n apeikonÐsewn,

TA = E ◦ L ◦ P ,

ìpou P : Rn → R(AT ) eÐnai h probol sto q¸ro gramm¸n, L : R(AT ) → R(A) eÐnai h

antistrèyimh apeikìnish L(x) = Ax, kai E : R(A)→ Rm eÐnai o egkleismìc E(y) = y.

T¸ra blèpoume oti se genikoÔc dianusmatikoÔc q¸rouc, qwrÐc epilegmènec b�seic, èqoume

mia an�logh paragontopoÐhsh, ìpou th jèsh tou q¸rou gramm¸n katalamb�nei to phlÐko.


Prìtash 2.16 E�n L : V → W eÐnai grammik apeikìnish, tìte L = E ◦ L ◦ P , ìpouP : V → V/ ker(L) eÐnai h kanonik epeikìnish, L : V/ ker(L) → im (L) eÐnai o kanonikìc

isomorfismìc kai E : im (L)→W eÐnai o egkleismìc.

�


Ask seic 3

Kef�laio 3

Nìrma kai eswterikì ginìmeno

82

Kef�laio 3 Nìrma kai eswterikì ginìmeno 83

Di�lexh 14

M koc dianÔsmatoc ston Rn

Sto epÐpedo, R2, brÐskoume to m koc ||x|| enìc dianÔsmatoc x = (x1, x2) qrhsimopoi¸ntac

to Pujagìreio je¸rhma:

||x||2 = x21 + x22 .

Sto q¸ro R3, efarmìzoume to Pujagìreio je¸rhma 2 forèc: e�n x = (x1, x2, x3) kai

u = (x1, x2, 0)

||x||2 = ||u||2 + x23

= x21 + x22 + x23 .

Qrhsimopoi¸ntac to sumbolismì tou an�strofou, autì gr�fetai

||x||2 = xT x .

Kat' analogÐa, orÐzoume to m koc ||x|| enìc dianÔsmatoc sto Rn:

||x||2 = x21 + x22 + · · ·+ x2n

= xT x .

Par�deigma 3.1 To m koc tou dianÔsmatoc x = (1, 2, −3) eÐnai√

14:

‖x‖2 = xTx = [1 2 − 3]

1

2

−3

= 12 + 22 + (−3)2 = 14 .

Orjog¸nia dianÔsmata ston Rn

Ektìc apì ta m kh, jèloume na metr�me kai gwnÐec metaxÔ dianusm�twn. Argìtera ja

mil soume gia ìlec tic gwnÐec, all� proc to parìn mac endiafèroun oi orjèc gwnÐec.

Pìte eÐnai dÔo dianÔsmata x , y orjog¸nia?

To Pujagìreio je¸rhma isqÔei kai antÐstrofa: èna trÐgwno eÐnai orjog¸nio mìnon ìtan

to tetr�gwno thc upoteÐnousac eÐnai Ðso me to �jroisma twn tetrag¸nwn twn 2 pleur¸n.

MporoÔme na ergastoÔme sto Rn, all� sthn pragmatikìthta oi metr seic ja eÐnai mèsa sto

epÐpedo pou perièqei to trÐgwno, dhlad mèsa sto dianusmatikì upìqwro pou par�getai apì

ta dianÔsmata x kai y. H gwnÐa ∠(x, y) eÐnai orj e�n kai mìnon e�n

||x||2 + ||y||2 = ||x− y||2 ,


dhlad e�n kai mìnon e�n

x1y1 + · · ·+ xnyn = 0

xT y = 0 .

H posìthta xT y genikeÔei stouc q¸rouc Rn to eswterikì ginìmeno dÔo dianusm�twn pou

gnwrÐzoume apì thn Analutik GewmetrÐa.

Orismìc 3.1. DÔo dianÔsmata x, y tou Rn lègontai orjog¸nia e�n to eswterikì

touc ginìmeno xT y eÐnai 0.

Par�deigma 3.2 To di�nusma x = (2, 2, −1) eÐnai orjog¸nio sto y = (−1, 2, 2):

xT y = [2 2 − 1]

−1

2

2

= 0 .

'Ena di�nusma eÐnai orjog¸nio ston eautì tou mìnon e�n èqei mhdenikì m koc: xTx = 0.

To monadikì tètoio di�nusma tou Rn eÐnai to 0.

Prìtash 3.1 E�n ta dianÔsmata v1, . . . , vk eÐnai mh mhdenik� kai orjog¸nia metaxÔ touc,

tìte eÐnai grammik� anex�rthta.

Apìdeixh. 'Estw ènac grammikìc sunduasmìc c1v1 + · · · + ckvk = 0. PaÐrnoume to

eswterikì ginìmeno me to v1:

vT1 (c1v1 + · · ·+ ckvk) = vT1 0 = 0

All� vT1 vi = 0 gia k�je i 6= 1, �ra èqoume

vT1 c1v1 = c1||v1||2 = 0

kai efìson ||v1|| 6= 0, èqoume c1 = 0

Parìmoia, ci = 0 gia k�je i, kai sumperaÐnoume oti ta dianÔsmata eÐnai grammik� anex�r-

thta.

�

EÐnai profanèc oti den isqÔei to antÐstrofo: duo grammik� anex�rthta dianÔsmata den

eÐnai upoqrewtik� orjog¸nia.

Drasthriìthta 3.1 BreÐte ta m kh kai to eswterikì ginìmeno twn x =

(1, 4, 0, 2) kai y = (2, −2, 1, 3).


Drasthriìthta 3.2 PoÐa zeÔgh apì ta dianÔsmata u1, u2, u3, u4 eÐnai orjog¸-

nia?

u1 =

1

2

−2

1

, u2 =

4

0

4

0

, u3 =

1

−1

−1

−1

, u4 =

1

1

1

1

.

Orjog¸nioi upìqwroi ston Rn

Ston R3, mÐa eujeÐa eÐnai k�jeth se èna epÐpedo ìtan sqhmatÐzei orj gwnÐa me k�je eujeÐa

tou epipèdou pou thn tèmnei.

An�loga, dÔo upìqwroi V kaiW tou q¸rou Rn eÐnai orjog¸nioi ìtan k�je di�nusma

tou V eÐnai orjog¸nio se k�je di�nusma tou W .

ParathroÔme oti dÔo epÐpeda W1 kai W2 sto R3 pou sqhmatÐzoun orj dÐedrh gwnÐa

den ikanopoioÔn aut th sunj kh. Pr�gmati, ac jewr soume mÐa b�sh apì dÔo orjog¸nia

dianÔsmata se k�je epÐpedo, u1, v1 sto W1 , u2, v2 sto W2. E�n ta W1 kai W2 tan

orjog¸nia, tìte ja eÐqame 4 dianÔsmata u1, v1, u2, v2 orjog¸nia metaxÔ touc. Apì thn

Prìtash 3.1 aut� ja tan grammik� anex�rthta. All� ston R3 den up�rqoun 4 grammik�

anex�rthta dianÔsmata.

Ja sumbolÐzoume thn orjogwniìthta dÔo grammik¸n upìqwrwn U kai V tou Rn me U⊥V .

Par�deigma 3.3 JewroÔme to epÐpedo V pou par�getai apì ta dianÔsmata v1 = (1, 0, 0, 0)

kai v2 = (1, 1, 0, 0). To di�nusma w = (0, 0, 4, 5) eÐnai orjog¸nio proc ta v1 kai v2. Su-

nep¸c h eujeÐa W pou par�getai apì to w eÐnai upìqwroc tou R4 orjog¸nioc proc ton V .

All� mèsa sto R4 up�rqei q¸roc gia akìmh ènan upìqwro orjog¸nio stouc V kai W : to

di�nusma z = (0, 0, 5, −4) eÐnai orjog¸nio proc ta v1, v2 kai w. H eujeÐa U pou par�getai

apì to z eÐnai orjog¸nia proc touc upìqwrouc V kai W :

U⊥V , U⊥W , V⊥W .

Drasthriìthta 3.3 DeÐxte oti oi upìqwroi tou R3, U = {(x, y, z, ∈ R3 :

x+ y = 0} kai V = 〈(1, 1, 0)〉 eÐnai orjog¸nioi.

Drasthriìthta 3.4 BreÐte a tètoio ¸ste oi upìqwroi tou R3, U = 〈(1, 1, a)〉kai V = 〈(1, a, 2)〉 eÐnai orjog¸nioi.


Prìtash 3.2 DÐdetai ènac m× n pÐnakac A. Tìte

1. Sto Rn o q¸roc gramm¸n tou A eÐnai orjog¸nioc sto mhdenìqwro tou A:

R(AT )⊥N (A)

2. Sto Rm o q¸roc sthl¸n tou A eÐnai orjog¸nioc ston aristerì mhdenìqwro tou A:

R(A)⊥N (AT ) .

Apìdeixh. ArkeÐ na deÐxoume thn pr¸th perÐptwsh, afoÔ h deÔterh prokÔptei exet�zontac

ton an�strofo pÐnaka AT .

JewroÔme èna x ∈ N (A) kai èna v ∈ R(AT ), kai jèloume na deÐxoume oti vTx = 0.

'Eqoume Ax = 0. Efìson v ∈ R(AT ), to v eÐnai grammikìc sunduasmìc twn gramm¸n

r1, . . . rm tou A,

v = z1r1 + · · ·+ zmrm ,

dhlad up�rqei z ∈ Rm tètoio ¸ste vT = zTA. 'Eqoume

vTx == (zTA)x = zT (Ax) = zT 0 = 0 .

�

Orjog¸nio sumpl rwma

UpenjumÐzoume oti oi diast�seic twn jemeliwd¸n upoq¸rwn enìcm×n pÐnaka A ikanopoioÔn

tic sqèseic:

dimR(AT ) + dimN (A) = n (3.1)

dimR(A) + dimN (AT ) = m (3.2)

Aut h parat rhsh upodeiknÔei oti o q¸roc gramm¸n kai o mhdenìqwroc den eÐnai dÔo opoioi-

d pote orjog¸nioi upìqwroi tou Rn: oi dÔo upìqwroi �gemÐzoun� ton Rn. Ac exet�soume pioprosektik� thn kat�stash. An W eÐnai to sÔnolo ìlwn twn dianusm�twn pou eÐnai orjog¸-

nia se ìla ta dianÔsmata tou q¸rou gramm¸n R(AT ), h Prìtash 3.2 lèei oti N (A) ⊆ W .

EÔkola ìmwc blèpoume oti isqÔei kai o antÐjetoc egkleismìc, W ⊆ N (A), dhlad o mhdenì-

qwroc perièqei k�je di�nusma pou eÐnai orjog¸nio se ìla ta dianÔsmata tou q¸rou gramm¸n.

Pr�gmati, e�n x ∈W tìte to x eÐnai orjog¸nio se k�je gramm tou A kai Ax = 0. Aut h

kat�stash parousi�zei arketì endiafèron ¸ste na thc d¸soume èna ìnoma:


Orismìc 3.2. JewroÔme grammikì upìqwro V tou Rn. To sÔnolo ìlwn twn dia-

nusm�twn tou Rn pou eÐnai orjog¸nia se k�je di�nusma tou V onom�zetai orjog¸nio

sumpl rwma tou V ston Rn, kai sumbolÐzetai V ⊥:

V ⊥ = {w ∈ Rn : wT v = 0 gia k�je v ∈ V } .

L mma 3.3 To orjog¸nio sumpl rwma V ⊥ eÐnai grammikìc upìqwroc tou Rn.

Apìdeixh. Prèpei na deÐxoume oti V ⊥ 6= ∅ kai oti gia k�je w1, w2 ∈ V ⊥ kai a ∈ R,aw1 + w2 ∈ V ⊥. Profan¸c 0 ∈ V ⊥, �ra V ⊥ 6= ∅. E�n gia k�je v ∈ V isqÔoun wT1 v = 0

kai wT2 v = 0, tìte (aw1 + w2)T v = awT1 v + wT2 v = 0.

�

Drasthriìthta 3.5 DeÐxte oti o upìqwroc U thc Drasthriìthtac 3.3 eÐnai to

orjog¸nio sumpl rwma tou upìqwrou V .

'Eqoume deÐxei oti o mhdenìqwroc eÐnai to orjog¸nio sumpl rwma tou q¸rou gramm¸n:

N (A) = R(AT )⊥ .

Ja deÐxoume oti kai o q¸roc gramm¸n eÐnai to orjog¸nio sumpl rwma tou mhdenìqwrou:

R(AT ) = N (A)⊥ .

H Prìtash 3.2 lèei oti R(AT ) ⊆ N (A)⊥. Gia na deÐxoume ton antÐjeto egkleismì jewroÔme

èna di�nusma z orjog¸nio sto N (A). 'Estw A′ o pÐnakac pou prokÔptei apì ton A episu-

n�ptontac wc mÐa epÐ plèon gramm th zT . O A′ èqei ton Ðdio mhdenìqwro me ton A, afoÔ h

nèa exÐswsh zTx = 0 ikanopoieÐtai gia k�je x ∈ N (A). EpÐshc èqei ton Ðdio arijmì sthl¸n,

n. SugkrÐnontac th sqèsh

dimR(A′T ) + dimN (A′) = n

me thn 3.1, kai afoÔ N (A′) = N (A), sumperaÐnoume oti dimR(A′ T ) = dimR(AT ). All�

autì shmaÐnei oti to di�nusma z exart�tai grammik� apì ta dianÔsmata miac b�shc touR(AT ),

dhlad oti an kei stoR(AT ). 'Eqoume apodeÐxei to pr¸to mèroc tou akìloujou jewr matoc.


Je¸rhma 3.4 DÐdetai ènac m× n pÐnakac A. Tìte

1. O mhdenìqwroc N (A) eÐnai to orjog¸nio sumpl rwma tou q¸rou gramm¸n R(AT )

ston Rn, kai o q¸roc gramm¸n eÐnai to orjog¸nio sumpl rwma tou mhdenìqwrou

ston Rn,N (A) = R(AT )⊥ kai R(AT ) = N (A)⊥ .

2. O aristerìc mhdenìqwroc N (AT ) eÐnai to orjog¸nio sumpl rwma tou q¸rou sthl¸n

R(A) ston Rm, kai o q¸roc sthl¸n eÐnai to orjog¸nio sumpl rwma tou aristeroÔ

mhdenìqwrou ston Rm,

N (AT ) = R(A)⊥ kai R(A) = N (AT )⊥ .

3.

Rn = R(AT )⊕N (A) , Rm = R(A)⊕N (AT ) .

Apìdeixh. 'Eqoume apodeÐxei to pr¸to mèroc tou jewr matoc. To deÔtero mèroc apodei-

knÔetai jewr¸ntac ton an�strofo pÐnaka. Gia to trÐto mèroc, arkeÐ na deÐxoume oti k�je

di�nusma x ∈ Rn, gr�fetai me monadikì trìpo wc �jroisma enìc dianÔsmatoc tou R(AT ) kai

enìc dianÔsmatoc tou N (A).

Pr¸ta deÐqnoume oti R(AT ) ∩ N (A) = {0}. E�n x ∈ R(AT ) ∩ N (A) tìte x eÐnai

orjog¸nio proc ton eautì tou, sunep¸c x = 0.

E�n {v1, . . . , vr} eÐnai b�sh tou R(AT ) kai {w1, . . . , wn−r} b�sh tou N (A), tìte to

sÔnolo {v1, . . . , vr, w1, . . . , wn−r} èqei n stoiqeÐa. E�n deÐxoume oti eÐnai grammik� anex�r-

thto, tìte eÐnai b�sh tou Rn. Upojètoume oti a1v1+ · · ·+arvr+b1w1+ · · ·+bn−rwn−r) = 0.

Jèloume na deÐxoume oti tìte ìla ta ai kai bj eÐnai 0. 'Eqoume

a1v1 + · · ·+ arvr = y = −(b1w1 + · · ·+ bn−rwn−r) .

All� h arister pleur� an kei sto R(AT ), h dexi� pleur� an kei sto N (A). 'Ara to

di�nusma y an kei sthn tom twn dÔo upoq¸rwn, kai sunep¸c y eÐnai to mhdenikì di�nusma

0 ∈ Rn. 'Ara a1v1 + · · · + arvr = 0, kai afoÔ v1, . . . , vr eÐnai grammik� anex�rthta, ìla ta

ai eÐnai 0. Parìmoia, b1w1 + · · ·+ bn−rwn−r = 0, kai afoÔ to w1, . . . , wn−r eÐnai grammik�

anex�rthta, ìla ta bj eÐnai 0.

AfoÔ {v1, . . . , vr, w1, . . . , wn−r} eÐnai b�sh tou Rn, k�je di�nusma tou Rn gr�fetai wc

�jroisma enìc stoiqeÐou tou R(AT ) kai enìc stoiqeÐou tou N (A). AfoÔ h tom twn dÔo

upoq¸rwn eÐnai {0},Rn = R(AT )⊕N (A) .

�

Me autì to Je¸rhma oloklhr¸netai h perigraf twn tess�rwn jemeliwd¸n upoq¸rwn

enìc pÐnaka, oi opoÐoi apoteloÔn dÔo zeÔgh orjogwnÐwn sumplhrwm�twn.


H dr�sh tou pÐnaka A.

T¸ra mporoÔme na oloklhr¸soume thn perigraf thc grammik c apeikìnishc TA pou polla-

plasi�zei k�je di�nusma tou Rn me ton m× n pÐnaka A.

E�n x ∈ N (A), tìte TA(x) = 0.

E�n to x eÐnai orjog¸nio sto mhdenoq¸ro, tìte x ∈ R(AT ), kai TA(x) ∈ R(A). All�

to shmantikì eÐnai oti aut h apeikìnish, apì to q¸ro gramm¸n R(AT ) sto q¸ro sthl¸n

R(A), eÐnai amfimonos manth.

Prìtash 3.5 Gia k�je di�nusma y ∈ R(A), up�rqei èna, kai mìnon èna, di�nusma x ∈R(AT ), tètoio ¸ste Ax = y.

Apìdeixh. AfoÔ to y an kei sto q¸ro sthl¸n, up�rqei u ∈ Rn tètoio ¸ste Au = y.

Apì thn Prìtash 3.11, up�rqoun monadik� dianÔsmata x ∈ R(AT ) kai w ∈ N (A), tètoia

¸ste u = x+w. All� Aw = 0, �ra Ax = Au = y. E�n up�rqei �llo di�nusma x′ ∈ R(AT )

me Ax′ = y, tìte x−x′ ∈ N (A). All� afoÔ R(AT ) eÐnai dianusmatikìc upìqwroc, x−x′ ∈R(AT ). Sunep¸c x− x′ = 0, kai èqoume monadikìthta.

�

E�n u ∈ Rn gr�fetai wc u = x + w, me x ∈ R(AT ) kai w ∈ N (A), orÐzoume P :

Rn −→ R(AT ) me P (u) = x kai E : R(A) −→ Rm me E(y) = y. SumbolÐzoume L :

R(AT ) −→ R(A) ton isomorfismì apì to q¸ro gramm¸n sto q¸ro sthl¸n tou A. Tìte h

apeikìnish TA paragontopoieÐtai wc sÔnjesh tou epimorfismoÔ P , tou isomorfismoÔ L kai

tou monomorfismoÔ E.

TA = E ◦ L ◦ P .

Anakefalai¸noume thn perigraf thc dr�shc tou pollaplasiasmoÔ me èna pÐnaka.

E�n A eÐnai ènac m × n pÐnakac t�xewc r, kai TA : Rn → Rm h grammik apeikìnish

x 7→ Ax, tìte

1. O q¸roc gramm¸n R(AT ) eÐnai upìqwroc tou Rn, di�stashc r.

2. O q¸roc sthl¸n R(A) eÐnai upìqwroc tou Rm, di�stashc r.

3. O mhdenoq¸roc N (A) eÐnai upìqwroc tou Rn di�stashc n− r.

4. O aristerìc mhdenìqwroc N (AT ) eÐnai upìqwroc tou Rm di�stashc m− r.

5. O mhdenoq¸roc eÐnai to orjog¸nio sumpl rwma tou q¸rou gramm¸n, N (A) = R(AT )⊥

kai Rn = R(AT )⊕N (A).

6. O aristerìc mhdenoq¸roc eÐnai to orjog¸nio sumpl rwma tou q¸rou sthl¸n,N (AT ) =

R(A)⊥ kai Rm = R(A)⊕N (AT ).

7. H grammik apeikìnish TA apeikonÐzei to dianusmatikì upìqwro R(AT ) tou Rn amfi-

monos manta sto dianusmatikì upìqwro R(A) tou Rm.


8. H grammik apeikìnish TAT apeikonÐzei to dianusmatikì upìqwro R(A) tou Rm amfi-

monos manta sto dianusmatikì upìqwro R(AT ) tou Rn.

9. H grammik apeikìnish TA paragontopoieÐtai wc sÔnjesh enìc epimorfismoÔ P : Rn −→R(AT ), enìc isomorfismoÔ L : R(AT ) −→ R(A) kai enìc monomorfismoÔE : R(A) −→Rm, TA = E ◦ L ◦ P .

Prosèxte oti oi dÔo amfimonos mantec apeikonÐseic sta 7 kai 8 den eÐnai upoqrewtik�

antÐstrofec h mÐa thc �llhc.

Aut h eikìna perigr�fetai parastatik� sto Sq ma 3.1.

Sq ma 3.1: H dr�sh tou m× n pÐnaka A.


'Askhsh 3.1 BreÐte èna di�nusma x orjog¸nio sto q¸ro gramm¸n tou A, èna

di�nusma orjog¸nio sto q¸ro sthl¸n, kai èna di�nusma orjog¸nio sto mhdenoq¸ro:

A =

1 2 1

2 4 3

3 6 4

.'Askhsh 3.2 BreÐte ìla ta dianÔsmata tou R3 pou eÐnai orjog¸nia sto (1, 1, 1)

kai sto (1, −1, 0).

'Askhsh 3.3 DÔo eujeÐec sto epÐpedo eÐnai orjog¸niec ìtan to ginìmeno twn klÐ-

sewn touc eÐnai −1. Efarmìste autì to krit rio stic eujeÐec pou par�gontai apì ta

dianÔsmata x = (x1, x2) kai y = (y1, y2), oi opoÐec èqoun klÐseic x2/x1 kai y2/y1,

gia na breÐte to krit rio orjogwniìthtac twn dianusm�twn, xT y = 0.

'Askhsh 3.4 P¸c gnwrÐzoume oti h i gramm enìc antistrèyimou pÐnaka B eÐnai

orjog¸nia sthn j st lh tou B−1, e�n i 6= j?

'Askhsh 3.5 DeÐxte oti to di�nusma x−y eÐnai orjog¸nio sto x+y e�n kai mìnon

e�n ||x|| = ||y||. Poi� idiìthta twn rìmbwn ekfr�zei autì to apotèlesma?

'Askhsh 3.6 BreÐte mÐa b�sh gia to orjog¸nio sumpl rwma tou q¸rou gramm¸n

tou A:

A =

[1 0 2

1 1 4

].

DiaqwrÐste to x = (3, 3, 3) se mÐa sunist¸sa sto q¸ro gramm¸n, kai se mÐa suni-

st¸sa sto mhdenoq¸ro tou A.

'Askhsh 3.7 Jewr ste ton upoq¸ro S tou R4 pou perièqei ìla ta dianÔsmata pou

ikanopoioÔn thn x1 +x2 +x3 +x4 = 0. BreÐte mÐa b�sh gia to q¸ro S⊥, pou perièqei

ìla ta dianÔsmata pou eÐnai orjog¸nia ston S.

'Askhsh 3.8 Gia na breÐte to orjog¸nio sumpl rwma tou epipèdou pou par�getai

apì ta dianÔsmata (1, 1, 2) kai (1, 2, 3), jewr ste aut� ta dianÔsmata wc grammèc

tou pÐnaka A, kai lÔste thn exÐswsh Ax = 0. JumhjeÐte oti to sumpl rwma eÐnai

olìklhrh eujeÐa.

'Askhsh 3.9 E�n V kai W eÐnai orjog¸nioi upìqwroi, deÐxte oti to mìno koinì

di�nusma eÐnai to mhdenikì: V ∩W = {0}.

'Askhsh 3.10 BreÐte ènan pÐnaka tou opoÐou o q¸roc gramm¸n perièqei to (1, 2, 1)

kai o mhdenoq¸roc perièqei to (1, −2, 1), deÐxte oti den up�rqei tètoioc pÐnakac.


'Askhsh 3.11 Kataskeu�ste mÐa omogen exÐswsh se treÐc agn¸stouc, thc opoÐ-

ac oi lÔseic eÐnai oi grammikoÐ sunduasmoÐ twn dianusm�twn (1, 1, 2) kai (1, 2, 3).

Autì eÐnai to antÐstrofo thc prohgoÔmenhc �skhshc, all� ta dÔo probl mata eÐnai

ousiastik� ta Ðdia.

'Askhsh 3.12 Sqedi�ste sto epÐpedo touc tèssereic jemeli¸deic upìqwrouc twn

pin�kwn

A =

[1 2

3 6

]kai B =

[1 0

3 0

].

'Askhsh 3.13 Sqedi�ste touc tèssereic jemeli¸deic upìqwrouc tou A, kai breÐte

tic sunist¸sec tou x sto q¸ro gramm¸n kai sto mhdenoq¸ro tou A, ìpou

A =

1 −1

0 0

0 0

kai x =

[2

0

].

'Askhsh 3.14 Se k�je perÐptwsh, kataskeu�ste ènan pÐnaka A me th zhtoÔmenh

idiìthta exhg ste giatÐ autì den eÐnai dunatì

1. O q¸roc sthl¸n perièqei ta dianÔsmata (1, 2, −3) kai (2, −3, 5), kai o mhde-

noq¸roc perièqei to (1, 1, 1).

2. O q¸roc gramm¸n perièqei ta (1, 2, −3) kai (2, −3, 5) kai o mhdenoq¸roc

perièqei to (1, 1, 1).

3. H exÐswsh Ax =

1

1

1

èqei lÔsh, kai AT

1

0

0

=

0

0

0

.4. To �jroisma twn sthl¸n eÐnai to di�nusma (0, 0, 0), kai to �jroisma twn gram-

m¸n eÐnai to di�nusma (1, 1, 1).

'Askhsh 3.15 Upojèste oti o upìqwroc S par�getai apì ta dianÔsmata

(1, 2, 2, 3) kai (1, 3, 3, 2). BreÐte dÔo dianÔsmata pou par�goun ton upìqwro S⊥.

Autì isodunameÐ me to na lÔsete thn exÐswsh Ax = 0 gia k�poio pÐnaka A. Poiìc

eÐnai o A?

'Askhsh 3.16 DeÐxte oti e�n o upìqwroc S perièqetai ston upìqwro V , tìte o

S⊥ perièqei ton V ⊥.


'Askhsh 3.17 BreÐte to orjog¸nio sumpl rwma S⊥ ìtan

1. S eÐnai o mhdenikìc upìqwroc tou R3.

2. S eÐnai o upìqwroc pou par�getai apì to (1, 1, 1).

3. S eÐnai o upìqwroc pou par�getai apì ta (2, 0, 0) kai (0, 0, 3).

'Askhsh 3.18 Kataskeu�ste ènan 3 × 3 pÐnaka A, qwrÐc mhdenik� stoiqeÐa, tou

opoÐou oi st lec eÐnai an� dÔo k�jetec. UpologÐste to ginìmeno ATA. GiatÐ eÐnai to

ginìmeno diag¸nioc pÐnakac?

'Askhsh 3.19 BreÐte ènan pÐnaka pou perièqei to di�nusma u = (1, 2, 3) sto q¸ro

gramm¸n kai sto q¸ro sthl¸n. BreÐte ènan �llo pÐnaka pou perièqei to u sto

mhdenoq¸ro kai sto q¸ro sthl¸n. Se poi� zeÔgh upoq¸rwn enìc pÐnaka den mporeÐ

na perièqetai to u?

'Askhsh 3.20 DeÐxte oti

V ⊥ = {w ∈ Rn : ∀v ∈ V, wT v = 0}

eÐnai pr�gmati grammikìc upìqwroc tou Rn, dhlad oti eÐnai èna mh kenì uposÔnolo

tou Rn, kleistì wc proc grammikoÔc sunduasmoÔc.

'Askhsh 3.21 DeÐxte oti e�n V eÐnai grammikìc upìqwroc tou Rn kai W = V ⊥,

tìte W⊥ = V , dhlad oti e�n o W eÐnai to orjog¸nio sumpl rwma tou V , tìte kai

o V eÐnai to orjog¸nio sumpl rwma tou W .

'Askhsh 3.22 ApodeÐxte oti h exÐswsh Ax = b èqei lÔsh e�n kai mìnon e�n

yT b = 0 gia k�je y pou ikanopoieÐ yTA = 0.


Di�lexh 15

Bèltistec lÔseic kai Probolèc

Epistrèfoume akìmh mÐa for� sthn exÐswsh Ax = b. 'Eqoume dei oti h exÐswsh èqei lÔseic

mìnon ìtan to di�nusma b an kei sto q¸ro sthl¸n tou pÐnaka A. Suqn� ìmwc jèloume na

broÔme thn kalÔterh dunat lÔsh thc exÐswshc, akìmh kai ìtan to b den an kei ston R(A).

Autì sumbaÐnei suqn� sthn an�lush peiramatik¸n dedomènwn, ìpou gia na periorÐsoume thn

pijanìthta tuqaÐou sf�lmatoc, paÐrnoume perissìterec metr seic. To apotèlesma eÐnai na è-

qoume èna sÔsthma me arket� perissìterec exis¸seic par� agn¸stouc, ìpou den perimènoume

na up�rqei akrib c lÔsh.

E�n antikatast soume to b me èna di�nusma b′ tou q¸rou sthl¸n R(A) tìte h exÐswsh

Ax = b′ èqei lÔsh. MporoÔme na broÔme mia bèltisth lÔsh thc exÐswshc, e�n antikatast -

soume to di�nusma b me to di�nusma tou q¸rou sthl¸n tou A pou eÐnai plhsièstero sto b

apì k�je �llo di�nusma tou q¸rou sthl¸n. Autì to di�nusma eÐnai h orjog¸nia probol

tou b sto q¸ro sthl¸n.

E�n sumbolÐsoume p thn orjog¸nia probol tou b sto q¸ro sthl¸n, èqoume mÐa nèa

exÐswsh

Ax = p.

Oi lÔseic aut c thc exÐswshc onom�zontai bèltistec lÔseic elaqÐstwn tetrag¸nwn thc arqi-

k c exÐswshc Ax = b, (deÐte thn 'Askhsh 3.23).

Par�deigma 3.4 Upojètoume oti melet�me thn ex�rthsh mÐac posìthtac b apì mÐa po-

sìthta a, kai anamènoume oti h b eÐnai an�logh proc thn a. Jèloume na broÔme ton stajerì

lìgo λ gia ton opoÐo

b = λa .

Upojètoume oti oi peiramatikèc metr seic dÐdoun tic timèc b1 gia a = 2, b2 gia a = 3 kai b3

gia a = 4. Gia na broÔme to λ jewroÔme treÐc exis¸seic me èna �gnwsto.

2x = b1

3x = b2

4x = b3 .

'Omwc autì to sÔsthma èqei lÔsh mìno ìtan to di�nusma (b1, b2, b3) eÐnai èna pollapl�sio

tou (2, 3, 4). H exÐswsh 2

3

4

x =

b1

b2

b3

(3.3)

èqei lÔsh mìnon ìtan to (b1, b2, b3) an kei sto q¸ro sthl¸n. Gia k�je tim tou x orÐzoume

to sf�lma

ε = ‖ax− b‖ =√

(2x− b1)2 + (3x− b2)2 + (4x− b3)2 ,


to opoÐo mhdenÐzetai mìno ìtan x apoteleÐ lÔsh thc exÐswshc 3.3. Sthn perÐptwsh pou h

exÐswsh 3.3 den èqei lÔsh, jewroÔme thn bèltisth lÔsh, thn tim tou x h opoÐa k�nei to

sf�lma ε ìso to dunatìn mikrìtero. Autì sumbaÐnei ìtan to di�nusma ax eÐnai Ðso me thn

orjog¸nia probol tou dianÔsmatoc b sto q¸ro sthl¸n, dhlad ìtan ax − b eÐnai k�jetosto a.

'Askhsh 3.23 UpologÐste thn par�gwgo ddx(ε2), kai deÐxte oti mhdenÐzetai akri-

b¸c ìtan ax− b eÐnai k�jeto sto a.

Probol se eujeÐa

Ac exet�soume pr¸ta thn probol se mÐa eujeÐa. JewroÔme ta dianÔsmata a kai b sto

epÐpedo. Sthn ��Eisagwg sth Grammik 'Algebra�� eÐdame (Kef�laio 4) thn probol tou

epipèdou R2 ston ϑ-�xona, dhlad sthn eujeÐa twn dianusm�twn pou eÐnai suggrammik� me

to (cosϑ, sinϑ). T¸ra jèloume na upologÐsoume thn probol enìc shmeÐou b tou Rn p�nw

sthn eujeÐa twn dianusm�twn pou eÐnai suggrammik� me to a ∈ Rn. To di�nusma probol c p

qarakthrÐzetai apì tic akìloujec idiìthtec:

1. To p eÐnai suggrammikì me to a, dhlad p = xa gia k�poio arijmì x ∈ R.

2. H diafor� b− p eÐnai orjog¸nia sto a, dhlad aT (b− p) = 0.

Apì autèc tic idiìthtec lamb�noume thn exÐswsh

aT (b− xa) = 0

thn opoÐa mporoÔme na lÔsoume gia na broÔme to x:

x =aT b

aTa.

Sunep¸c to di�nusma probol c eÐnai

p = xa =aT b

aTaa .

Jèloume na ekfr�soume thn probol wc mÐa grammik apeikìnish apì ton Rn ston Rn, hopoÐa apeikonÐzei ton Rn sthn eujeÐa V = {ta : t ∈ R}, kai na broÔme ton antÐstoiqo pÐnaka.

Ston prohgoÔmeno upologismì mporoÔme na antistrèyoume th di�taxh tou a kai tou x:

p = ax = aaT b

aTa,

kai na efarmìsoume thn prosetairistik idiìthta:

p =1

aTaaaT b .

Parathr ste oti aTa eÐnai jetikìc arijmìc, to tetr�gwno tou m kouc tou a, en¸ aaT eÐnai

tetragwnikìc pÐnakac.


Ton pÐnaka

P =1

aTaaaT

onom�zoume pÐnaka probol c. Gia na prob�loume to di�nusma b ∈ Rn sthn eujeÐa pou

orÐzei to di�nusma a, arkeÐ na to pollaplasi�soume me ton pÐnaka P .

Par�deigma 3.5 SuneqÐzoume to Par�deigma 3.4, me b = (4, 6, 9), dhlad jewroÔme to

sÔsthma 2

3

4

x =

4

6

9

.Autì den èqei lÔsh, afoÔ to di�nusma (4, 6, 9) den an kei sto q¸ro pou par�gei to (2, 3, 4).

H bèltisth lÔsh eÐnai x, tètoia ¸ste

[2 3 4

] 4

6

9

− x 2

3

4

= 0 ,

dhlad

x =(2, 3, 4) · (4, 6, 9)

22 + 32 + 42=

62

29.

SumperaÐnoume oti h bèltisth tim gia to λ pou prokÔptei apì ta 3 shmeÐa (2, 4), (3, 6) kai

(4, 9) eÐnai λ = 6229 .

'Askhsh 3.24 BreÐte thn probol tou dianÔsmatoc (7, 4) p�nw ston upìqwro

pou par�getai apì to di�nusma (1, 2).

'Askhsh 3.25 BreÐte ton pÐnaka probol c pou antistoiqeÐ sthn probol twn

dianusm�twn tou epipèdou R2 p�nw sthn eujeÐa 3x− 2y = 0.

'Askhsh 3.26 BreÐte ton pÐnaka probol c P1 sthn eujeÐa me dieÔjunsh a = (1, 3),

kaj¸c kai ton pÐnaka probol c P2 sthn eujeÐa pou eÐnai k�jeth sto a. UpologÐste

touc pÐnakec P1 + P2 kai P1P2. Exhg ste to apotèlesma.

'Askhsh 3.27 Ston q¸ro Rn, poi� gwnÐa sqhmatÐzei to di�nusma (1, 1, . . . , 1) me

touc �xonec suntetagmènwn? BreÐte ton pÐnaka probol c se autì to di�nusma.

'Askhsh 3.28 Poiì pollapl�sio tou a = (1, 1, 1) eÐnai plhsièstero sto shmeÐo

b = (2, 4, 4)? BreÐte epÐshc to shmeÐo sthn eujeÐa me dieÔjunsh b pou eÐnai plhsiè-

stero sto a.

'Askhsh 3.29 DeÐxte oti o pÐnakac probol c P = 1aT a

aaT eÐnai summetrikìc kai

ikanopoieÐ th sqèsh P 2 = P .


'Askhsh 3.30 Poioc pÐnakac P prob�lei k�je shmeÐo tou R3 sthn eujeÐa ìpou

tèmnontai ta epÐpeda x+ y + t = 0 kai x− t = 0?

'Askhsh 3.31 Gia ta akìlouja dianÔsmata, sqedi�ste sto kartesianì epÐpedo thn

probol tou b sto a, kai sth sunèqeia upologÐste thn probol , apì thn èkfrash

p = xa:

1. b =

[cosϑ

sinϑ

]kai a =

[1

0

]

2. b =

[1

1

]kai a =

[1

−1

]

'Askhsh 3.32 UpologÐste thn probol tou b sthn eujeÐa me dieÔjunsh a, kai

elègxte oti to dianusmatikì sf�lma e = b− p eÐnai orjog¸nio sto a:

1. b =

1

2

2

kai a =

1

1

1

2. b =

1

3

1

kai a =

−1

−3

−1

'Askhsh 3.33 'Estw a di�nusma tou Rn kai èstw P o pÐnakac probol c tou a.

DeÐxte oti to �jroisma twn diagwnÐwn stoiqeÐwn tou P isoÔtai me 1.

'Askhsh 3.34 'Estw a = (1, 2, −1, 3).

1. BreÐte ton pÐnaka probol c P sto di�nusma a.

2. BreÐte mia b�sh tou mhdenìqwrou N (P ).

3. BreÐte èna mh mhdenikì di�nusma v tou R4 tou opoÐou h probol sto a na eÐnai

to mhdenikì di�nusma.

Probol se upìqwro

JewroÔme t¸ra to prìblhma se perissìterec diast�seic. Jèloume na prob�loume to di�nu-

sma b se èna upìqwro V di�stashc k mèsa ston Rm. MporoÔme gia eukolÐa na upojèsoume

oti k = 2 kai m = 3, qwrÐc ousiastik diafor� sth diadikasÐa. JewroÔme loipìn dÔo dianÔ-

smata a1 kai a2 tou R3, ta opoÐa apoteloÔn b�sh tou V , kai ton m× k pÐnaka A me st lec

ta dianÔsmata ai, ètsi ¸ste V = R(A). AfoÔ h probol p brÐsketai sto q¸ro sthl¸n tou


A, èqoume

p = Ax

gia k�poio x ∈ Rk. AfoÔ h probol eÐnai orjog¸nia, to di�nusma b − Ax eÐnai orjog¸nio

sto q¸ro sthl¸n tou A, kai apì to Je¸rhma 3.4 an kei ston aristerì mhdenoq¸ro tou A:

AT (b−Ax) = 0 .

'Etsi èqoume thn exÐswsh

ATAx = AT b ,

h opoÐa ja mac d¸sei th bèltisth lÔsh x, apì thn opoÐa mporoÔme na upologÐsoume to p.

E�n o pÐnakac ATA eÐnai antistrèyimoc, èqoume

x = (ATA)−1AT b ,

kai h probol tou b ston upìqwro V = R(A) eÐnai

p = Ax = A(ATA)−1AT b .

O pÐnakac probol c eÐnai

P = A(ATA)−1AT .

Se aut thn èkfrash, A eÐnai m× k pÐnakac, opìte ATA eÐnai tetragwnikìc k × k pÐnakac,

kai P eÐnai m×m pÐnakac.

An sugkrÐnoume me thn perÐptwsh thc probol c se eujeÐa, ìpou k = 1, blèpoume oti o

m× 1 pÐnakac A eÐnai to di�nusma a, kai o antistrèyimoc k× k pÐnakac ATA eÐnai o jetikìc

arijmìc aTa, me antÐstrofo 1aT a

. Autìc metatÐjetai me ton pÐnaka A, kai sunep¸c mporoÔme

na gr�youme

a(aTa)−1aT =1

aTaaaT .

Ja deÐxoume oti h upìjesh pwc ATA eÐnai antistrèyimoc ikanopoieÐtai p�nta ìtan oi

st lec tou A eÐnai grammik� anex�rthtec, ìpwc sthn perÐptwsh pou apoteloÔn b�sh tou

upìqwrou V .

L mma 3.6 O pÐnakac ATA èqei ton Ðdio mhdenìqwro me ton A.

Apìdeixh. EÐnai profanèc oti e�n Ax = 0 tìte ATAx = 0, dhlad oti N (A) ⊆ N (ATA).

Gia na deÐxoume ton antÐstrofo egkleismì jewroÔme x tètoio ¸ste ATAx = 0, opìte

xT (ATAx) = 0.

All� xT (ATAx) = (xTAT )Ax = (Ax)TAx = ||Ax||2.'Ara to di�nusma Ax èqei mhdenikì m koc, kai sunep¸c Ax = 0, dhlad x ∈ N (A).

�

Prìtash 3.7 'Enac m×m pÐnakac P eÐnai pÐnakac probol c se èna upìqwro tou Rm e�n

kai mìnon e�n P eÐnai summetrikìc kai P 2 = P .


Apìdeixh. 'Estw V ènac upìqwroc tou Rm, kai A o pÐnakac me st lec ta dianÔsmata

mÐac b�shc tou V . Tìte o pÐnakac probol c ston upìqwro V eÐnai o P = A(ATA)−1AT .

EÔkola elègqoume oti P 2 = P ,

P 2 = A(ATA)−1ATA(ATA)−1AT

= A(ATA)−1AT

= P.

O an�strofoc tou P eÐnai o pÐnakac

P T = (A(ATA)−1AT )T

= (AT )T((ATA)−1

)TAT

= A((ATA)T

)−1AT

= A(AAT )−1AT

= P

Antistrìfwc, e�n om×m pÐnakac P ikanopoieÐ tic sqèseic P 2 = P kai P = P T , ja deÐxoume

oti P eÐnai o pÐnakac probol c sto q¸ro sthl¸n tou. Profan¸c, gia k�je b ∈ Rm, Pban kei sto q¸ro sthl¸n tou P . Gia na deÐxoume oti Pb eÐnai h probol tou b ston upìqwro

V = R(P ) arkeÐ na deÐxoume oti b− Pb eÐnai orjog¸nio ston V .

'Estw v di�nusma tou V . Tìte v eÐnai grammikìc sunduasmìc twn sthl¸n tou P , dhlad

up�rqei c ∈ Rm tètoio ¸ste v = Pc, kai èqoume

(b− Pb)T v = (b− Pb)TPc

= (bT − bTP T )Pc

= bT (I − P T )Pc

= bT (P − P TP )c.

All� P T = P kai P 2 = P , �ra P − P TP = P − P = 0.

�

Par�deigma 3.6 SuneqÐzoume to Par�deigma 3.4, me b = (4, 6, 9), all� t¸ra upojètoume

oti h sqèsh metaxÔ twn posot twn a kai b eÐnai

b = λa+ µ .

Me ta Ðdia dedomèna, (2, 4), (3, 6) kai (4, 9), èqoume treÐc exis¸seic me dÔo agn¸stouc gia

na broÔme ta λ kai µ:

2x1 + x2 = 4

3x1 + x2 = 6

4x1 + x2 = 9 ,


tic opoÐec gr�foume wc èna sÔsthma

Ax =

2 1

3 1

4 1

[ x1

x2

]=

4

6

9

.To di�nusma (4, 6, 9) den an kei sto q¸ro sthl¸n tou pÐnaka A, kai to sÔsthma den

èqei lÔsh. To sf�lma ε = ‖b − Ax‖ elaqistopoi tai gia thn tim x tou x = (x1, x2) gia

thn opoÐa to di�nusma b − Ax eÐnai orjog¸nio sto q¸ro sthl¸n tou A. 'Etsi èqoume thn

exÐswsh elaqÐstwn tetrag¸nwn:

AT (b−Ax) =

[2 3 4

1 1 1

] 4

6

9

− 2 1

3 1

4 1

[ x1

x2

] = 0 ,

dhlad [62

19

]−

[29 9

9 3

][x1

x2

]= 0

me lÔsh [x1

x2

]=

1

6

[3 −9

−9 29

][62

19

]=

[15676

].

'Ara h bèltisth eujeÐa pou kajorÐzetai apì ta shmeÐa (2, 4), (3, 6) kai (4, 9) èqei exÐswsh

6b = 15a+ 7 .

'Askhsh 3.35 BreÐte th bèltisth lÔsh elaqÐstwn tetrag¸nwn thc exÐswshc Ax =

b, kai upologÐste thn probol p = Ax, e�n

A =

1 0

0 1

1 1

kai b =

1

1

0

.EpalhjeÔste oti to dianusmatikì sf�lma e = b− p eÐnai orjog¸nio stic st lec tou

A.

'Askhsh 3.36 UpologÐste to tetr�gwno tou sf�lmatoc ε2 = ||Ax − b||2, kaibreÐte tic merikèc parag¸gouc tou ε2 wc proc u kai v, e�n

A =

1 0

0 1

1 1

, x =

[u

v

]kai b =

1

3

4

.Jèsate tic parag¸gouc Ðsec me mhdèn, kai sugkrÐnete me tic exis¸seic ATAx =

AT b, gia na deÐxete oti o logismìc kai h gewmetrÐa katal goun stic Ðdiec exis¸seic.

UpologÐste to x kai thn probol p = Ax. GiatÐ eÐnai p = b?


'Askhsh 3.37 BreÐte thn probol tou b = (4, 3, 1, 0) p�nw sto q¸ro sthl¸n

tou

A =

1 −2

1 −1

1 0

1 2

.

'Askhsh 3.38 BreÐte thn bèltisth lÔsh elaqÐstwn tetrag¸nwn x, tou sust matoc

exis¸sewn 3x = 10 kai 4x = 5. Poio eÐnai to tetr�gwno tou sf�lmatoc ε2 pou

elaqistopoieÐtai? EpalhjeÔste oti to dianusmatikì sf�lma e = (10 − 3x, 5 − 4x)

eÐnai orjog¸nio sth st lh (3, 4).

'Askhsh 3.39 BreÐte thn probol tou b sto q¸ro sthl¸n tou A:

A =

1 1

1 −1

−2 4

, b =

1

2

7

.Diaqwr ste to b se �jroisma p+ q, me p sto q¸ro sthl¸n tou A kai q orjog¸nio

proc autìn. Se poiì jemeli¸dh upìqwro tou A brÐsketai to di�nusma q?

'Askhsh 3.40 DeÐxte oti e�n o pÐnakac P ikanopoieÐ th sqèsh P = P TP , tìte P

eÐnai pÐnakac probol c. EÐnai o mhdenikìc pÐnakac P = 0 pÐnakac probol c, kai se

poio upìqwro?

'Askhsh 3.41 Ta dianÔsmata a1 = (1, 1, 0) kai a2 = (1, 1, 1) par�goun èna epÐ-

pedo sto R3. BreÐte ton pÐnaka probol c sto epÐpedo, kai èna mh mhdenikì di�nusma

b to opoÐo prob�letai sto 0.

'Askhsh 3.42 E�n V eÐnai o upìqwroc pou par�getai apì ta (1, 1, 0, 1) kai

(0, 0, 1, 0) breÐte

1. mÐa b�sh gia to orjog¸nio sumpl rwma V ⊥.

2. ton pÐnaka probol c P sto V .

3. to di�nusma sto V to opoÐo eÐnai plhsièstero proc to (0, 1, 0, −1) ∈ V ⊥

'Askhsh 3.43 E�n P eÐnai h probol sto q¸ro sthl¸n tou pÐnaka A, poi� eÐnai h

probol ston aristerì mhdenoq¸ro tou A?

'Askhsh 3.44 E�n Pσ = A(ATA)−1AT eÐnai o pÐnakac probol c sto q¸ro sthl¸n

tou A, poiìc eÐnai o pÐnakac probol c Pγ sto q¸ro gramm¸n tou A?


'Askhsh 3.45 JewroÔme ton dianusmatikì upìqwro V tou R4 pou par�getai apì

ta dianÔsmata

(1, 2, 0, 3), (2, 1, 1, 2) (−1, 4, −2, 5)

1. BreÐte to orjog¸nio sumpl rwma V ⊥ tou V .

2. Gr�yte to di�nusma x = (−4, 15, 7, 8) wc �jroisma x = v + w, ìpou v ∈ Vkai w ∈ V ⊥.


Di�lexh 16

Orjog¸nioi pÐnakec

JewroÔme èna dianusmatikì upìqwro V ⊆ Rn di�stashc dimV = k, kai mÐa b�sh v1, . . . , vk

tou V . Tìte k�je di�nusma u ∈ V mporeÐ na ekfrasteÐ wc grammikìc sunduasmìc

u = a1v1 + a2v2 + · · ·+ akvk.

To eswterikì ginìmeno twn dianusm�twn u kai w = b1v1 + b2v2 + · · ·+ bkvk eÐnai

uTw = (a1vT1 + · · · akvTk )(b1v1 + · · ·+ bkvk)

=k∑i=1

k∑j=1

aibjvTi vj .

E�n upojèsoume oti ta dianÔsmata thc b�shc eÐnai an� dÔo orjog¸nia metaxÔ touc, dhlad

oti vTi vj = 0 e�n i 6= j, to eswterikì ginìmeno uTw gÐnetai

uTw =k∑i=1

aibivTi vi

=k∑i=1

aibi||vi||2.

EÐnai fanerì oti mÐa b�sh apì orjog¸nia dianÔsmata mporeÐ na aplopoi sei shmantik� touc

upologismoÔc. Se mÐa tètoia b�sh, mìno mÐa akìmh beltÐwsh mporoÔme na k�noume: to m koc

k�je dianÔsmatoc thc b�shc na eÐnai ||vi||2 = 1. Tìte to eswterikì ginìmeno twn u kai w

lamb�nei thn aploÔsterh dunat morf :

uTw =k∑i=1

aibi

MÐa tètoia bèltisth b�sh thn onom�zoume orjokanonik .

Orismìc 3.3. Ta dianÔsmata q1, q2, . . . , qk eÐnai orjokanonik� e�n

qTi qj =

0 e�n i 6= j

1 e�n i = j

MÐa b�sh pou apoteleÐtai apì orjokanonik� dianÔsmata onom�zetai orjokanonik b�-

sh. 'Enac tetragwnikìc pÐnakac tou opoÐou oi st lec eÐnai orjokanonik� dianÔsmata ono-

m�zetai orjog¸nioc.

Prosèxte oti o ìroc orjog¸nioc qrhsimopoieÐtai mìno gia tetragwnikoÔc pÐnakec. 'Enac

mh tetragwnikìc pÐnakac me orjokanonikèc st lec den onom�zetai orjog¸nioc.


'Askhsh 3.46 DeÐxte oti e�n q1, . . . , qk eÐnai mÐa orjokanonik b�sh tou V , tìte

oi suntelestèc a1, . . . , ak tou dianÔsmatoc u = a1q1 + · · ·+ akqk eÐnai ai = qTi u.

To shmantikìtero par�deigma orjokanonik c b�shc eÐnai h kanonik b�sh e1, . . . , en tou

Rn. O orjog¸nioc pÐnakac pou èqei aut� ta dianÔsmata wc st lec, me th di�taxh e1, . . . , en

eÐnai o tautotikìc n×n pÐnakac I. Oi Ðdiec st lec me diaforetik di�taxh dÐdoun touc pÐnakec

met�jeshc, oi opoÐoi eÐnai epÐshc orjog¸nioi pÐnakec.

Prìtash 3.8 E�n o m× n pÐnakac M èqei orjokanonikèc st lec tìte MTM = I.

Eidikìtera, e�n Q eÐnai orjog¸nioc pÐnakac, tìte o an�strofoc pÐnakac eÐnai kai antÐstrofoc,

QT = Q−1 .

Apìdeixh. Upojètoume oti q1, . . . , qn ∈ Rm eÐnai oi st lec tou M . Tìte MTM eÐnai o

n× n pÐnakac me stoiqeÐo sth jèsh (i, j) to qTi qj . All�

qTi qj =

{0 e�n i 6= j

1 e�n i 6= j .

Sunep¸c MTM eÐnai o tautotikìc n× n pÐnakac kai MT eÐnai aristerì antÐstrofo tou M .

E�n o pÐnakac eÐnai tetragwnikìc kai èqei aristerì antÐstrofo, gnwrÐzoume oti eÐnai

antistrèyimoc. 'Ara QT eÐnai o antÐstrofoc pÐnakac.

�

Par�deigma 3.7 O pÐnakac peristrof c Q =

[cosϑ − sinϑ

sinϑ cosϑ

]eÐnai orjog¸nioc. O

Q peristrèfei kat� gwnÐa ϑ, en¸ o an�strofoc QT =

[cosϑ sinϑ

− sinϑ cosϑ

]peristrèfei kat�

gwnÐa −ϑ. Oi st lec eÐnai orjog¸niec, kai afoÔ sin2 ϑ+ cos2 ϑ = 1, èqoun m koc 1.

Par�deigma 3.8 'Opwc anafèrame prohgoumènwc, k�je pÐnakac met�jeshc eÐnai orjo-

g¸nioc. Eidikìtera o pÐnakac

P =

[0 1

1 0

],

pou parist�nei thn an�klash ston �xona x = y. Gewmetrik�, k�je orjog¸nioc pÐnakac eÐnai

sÔnjesh mÐac peristrof c kai mÐac an�klashc.

Oi orjog¸nioi pÐnakec èqoun akìma mÐa shmantik idiìthta:

Prìtash 3.9 O pollaplasiasmìc me èna orjog¸nio pÐnaka Q af nei to m koc amet�blh-

to: gia k�je x ∈ Rn,||Qx|| = ||x||.

Genikìtera, pollaplasiasmìc me orjog¸nio pÐnaka af nei to eswterikì ginìmeno amet�blh-

to: gia k�je x, y ∈ Rn,(Qx)T (Qy) = xT y.


Apìdeixh. GnwrÐzoume oti (Qx)T = xTQT . All� QTQ = I kai èqoume:

(Qx)T (Qy) = xTQTQy = xT Iy = xT y.

�

E�n o pÐnakac Q eÐnai orjog¸nioc, tìte QT = Q−1, kai sunep¸c QQT = I. Autì shmaÐnei

oti oi grammèc enìc orjog¸niou pÐnaka eÐnai epÐshc orjokanonik� dianÔsmata.

OrjokanonikopoÐhsh Gram-Schmidt

Ja deÐxoume oti e�n v1, . . . , vk eÐnai opoiad pote b�sh tou upoq¸rou V ⊆ Rn, mporoÔ-me na kataskeu�soume apì thn v1, . . . , vk mÐa orjokanonik b�sh q1, . . . , qk, tètoia ¸ste

gia k�je j = 1, . . . , k, ta dianÔsmata q1, . . . , qj par�goun ton Ðdio upìqwro pou par�goun

ta dianÔsmata v1, . . . , vj . Aut h diadikasÐa onom�zetai orjokanonikopoÐhsh Gram-

Schmidt. Ja thn perigr�youme sthn perÐptwsh tri¸n dianusm�twn v1, v2, v3. Upojètoume

oti ta v1, v2, v3 eÐnai grammik� anex�rthta. Arqik� ja ta antikatast soume me trÐa orjo-

g¸nia dianÔsmata w1, w2, w3. Sth sunèqeia, diairoÔme k�je di�nusma wi me to m koc tou

kai èqoume ta orjokanonik� dianÔsmata q1, q2, q3.

Jètoume w1 = v1. Jèloume w2 orjog¸nio sto w1 kai tètoio ¸ste ta w1 kai w2 na

par�goun ton Ðdio upìqwro pou par�goun ta v1 kai v2. AfairoÔme apì to v2 thn probol

tou sthn eujeÐa pou par�getai apì to w1.

w2 = v2 −wT1 v2

wT1 w1w1 .

Elègqoume oti w1 kai w2 eÐnai orjog¸nia:

wT1 w2 = wT1 v2 −wT1 v2

wT1 w1wT1 w1

= 0

To v3 den perièqetai sto epÐpedo pou par�goun ta w1, w2, afoÔ upojèsame oti ta v1, v2

kai v3 eÐnai grammik� anex�rthta. Gia na broÔme to w3 ja afairèsoume thn probol tou v3

sto epÐpedo pou par�goun ta v1 kai v2. 'Estw A o pÐnakac me st lec ta dianÔsmata w1 kai

w2. Tìte

ATA =

[wT1 w1 0

0 wT2 w2

]kai, qrhsimopoi¸ntac to sumbolismì twn mplok, A = [w1w2], h probol tou v3 ston upì-

qwro pou par�getai apì ta w1 kai W2 eÐnai

A(ATA)−1AT v3 = [w1w2]

wT1 w1 0

0 wT2 w2

−1 [

wT1wT2

]v3


=w1

wT1 w1wT1 v3 +

w2

wT2 w2wT2 v3

=wT1 v3

wT1 w1w1 +

wT2 v3

wT2 w2w2

ParathroÔme oti, epeid ta dianÔsmata w1 kai w2 eÐnai orjog¸nia, h probol sto epÐpedo

pou par�goun ta w1 kai w2 eÐnai to �jroisma twn probol¸n stic eujeÐec twn w1 kai w2.

Katal goume pwc

w3 = v3 −wT1 v3||w1||2

w1 −wT2 v3||w2||2

w2.

Ta dianÔsmata w1, w2, w3 eÐnai t¸ra orjog¸nia. Gia na broÔme thn orjokanonik b�sh tou

V arkeÐ na diairèsoume k�je di�nusma me to m koc tou,

q1 =w1

||w1||, q2 =

w2

||w2||, q3 =

w3

||w3||.

Par�deigma 3.9 JewroÔme ta dianÔsmata

v1 =

1

0

1

0

, v2 =

1

0

0

1

, v3 =

2

1

0

0

.Tìte

w1 =

1

0

1

0

,

w2 =

1

0

0

1

− 1

2

1

0

1

0

=

12

0

−12

1

,

w3 =

2

1

0

0

− 2

2

1

0

1

0

− 1

3/2

12

0

−12

1

=

23

1

−23

−23

.'Ara

q1 =1√2

1

0

1

0

, q2 =

√2√3

12

0

−12

1

=

1√6

0

− 1√6√2√3

, q3 =3√21

23

1

−23

−23

=

2√213√21

− 2√21

− 2√21

.


'Opwc perigr�yame th diadikasÐa thc apaloif c Gauss mèsw thc paragontopoÐhshc A =

LU , mporoÔme na perigr�youme kai thn orjokanonikopoÐhsh Gram-Schmidt mèsw mÐac pa-

ragontopoÐhshc tou pÐnaka A o opoÐoc èqei wc st lec ta dianÔsmata v1, v2, . . . , vk:

A = QR,

ìpou Q eÐnai o pÐnakac me orjokanonikèc st lec q1, q2, . . . , qk, kai R eÐnai o pÐnakac pou

antistrèfei th diadikasÐa Gram-Schmidt. AfoÔ ta q1, . . . , qk eÐnai orjog¸nia kai vj ∈〈q1, . . . , qj〉,

v1 = c11q1 = (qT1 v1)q1

v2 = c12q1 + c22q2 = (qT1 v2)q1 + (qT2 v2)q2

· · · = · · ·

vk = c1kq1 + · · ·+ ckkqk = (qT1 vk)q1 + · · ·+ (qTk vk)qk .

Blèpoume oti e�n A eÐnai m× k pÐnakac, R eÐnai o �nw trigwnikìc k× k pÐnakac me stoiqeÐo

sth jèsh i, j

Rij = qTi vj gia j ≥ i .

'Eqoume apodeÐxei thn akìloujh Prìtash.

Prìtash 3.10 K�je m× k pÐnakac A me grammik� anex�rthtec st lec mporeÐ na para-

gontopoihjeÐ sth morf

A = QR ,

ìpou o Q eÐnai m× k pÐnakac me orjokanonikèc st lec, kai o R eÐnai k × k �nw trigwnikìc

kai antistrèyimoc. E�n m = k, tìte Q eÐnai orjog¸nioc pÐnakac.

Epistrèfontac sto Par�deigma, o pÐnakac pou èqei wc st lec ta dianÔsmata v1, v2, v3

paragontopoieÐtai wc1 1 2

0 0 1

1 0 0

0 1 0

= QR =

1√2

1√6

2√21

0 0 3√21

1√2−1√6

−2√21

0 2√6

−2√21

√

2 1√2

√2

0 3√6

2√6

0 0 7√21

.

H paragontopoÐhsh A = QR aplopoieÐ to prìblhma eÔreshc bèltisthc lÔshc elaqÐstwn

tetrag¸nwn. H exÐswsh

ATAx = AT b

gÐnetai

RTQTQRx = RTQT b.

AfoÔ QTQ = I kai RT eÐnai antistrèyimoc, èqoume

Rx = QT b


kai afoÔ R eÐnai �nw trigwnikìc, to di�nusma x upologÐzetai me an�dromh antikat�stash.

'Askhsh 3.47 E�n u eÐnai monadiaÐo di�nusma, deÐxte oti Q = I − 2uuT eÐnai sum-

metrikìc orjog¸nioc pÐnakac. (EÐnai mÐa an�klash, kai onom�zetai metasqhmatismìc

Householder). UpologÐste ton Q ìtan uT =

[1

2

1

2− 1

2− 1

2

].

'Askhsh 3.48 Prob�lete to di�nusma b = (1, 2) se dÔo m orjog¸nia dianÔsmata,

a1 = (1, 0) kai a2 = (1, 1). EpalhjeÔste oti to �jroisma twn dÔo probol¸n den

eÐnai Ðso proc to b.

'Askhsh 3.49 DeÐxte oti ènac �nw trigwnikìc orjog¸nioc pÐnakac prèpei na eÐnai

diag¸nioc.

'Askhsh 3.50 Apì ta mh orjog¸nia dianÔsmata v1, v2, v3, breÐte orjog¸nia dia-

nÔsmata q1, q2, q3.

v1 =

1

1

0

, v2 =

1

0

1

, u3 =

0

1

1

.

'Askhsh 3.51 Poi� eÐnai ta dunat� dianÔsmata v1 kai v2, pou dÐnoun met� apì

orjokanonikopoÐhsh Gram-Schmidt ta dianÔsmata q1 kai q2.

'Askhsh 3.52 Poiì pollapl�sio tou a1 = (1, 1) prèpei na afairejeÐ apì to

a2 = (4, 0), ¸ste to apotèlesma na eÐnai orjog¸nio proc to a1. Paragontopoi ste

ton pÐnaka

[1 4

1 0

]se ginìmeno QR ìpou Q eÐnai orjog¸nioc.

'Askhsh 3.53 Efarmìste th diadikasÐa Gram-Schmidt sta dianÔsmata

v1 =

0

0

1

, v2 =

0

1

1

, u3 =

1

1

1

,kai ekfr�ste to apotèlesma sth morf A = QR.

'Askhsh 3.54 E�n A = QR, ìpou oi st lec tou Q eÐnai orjog¸nia dianÔsmata,

breÐte ènan aplì tÔpo gia ton pÐnaka probol c sto q¸ro sthl¸n tou A.


'Askhsh 3.55 BreÐte trÐa orjokanonik� dianÔsmata q1, q2, q3 ∈ R3, tètoia ¸ste

ta q1, q2 na par�goun to q¸ro sthl¸n tou pÐnaka

A =

1 1

2 −1

−2 4

.Poiìc jemeli¸dhc upìqwroc tou A perièqei to di�nusma q3? BreÐte th bèltisth lÔsh

elaqÐstwn tetrag¸nwn thc exÐswshc Ax = b, ìtan bT = [1 2 7].

'Askhsh 3.56 Me ton pÐnaka A thc 'Askhshc 3.55, kai to di�nusma b = [1 1 1]T ,

qrhsimopoi ste thn paragontopoÐhsh A = QR gia na lÔsete to prìblhma elaqÐstwn

tetrag¸nwn Ax = b.

'Askhsh 3.57 Efarmìste th diadikasÐa Gram-Schmidt sta dianÔsmata

(1, −1, 0), (0, 1, −1) kai (1, 0, −1) gia na breÐte orjokanonik b�sh tou epipè-

dou x1 + x2 + x3 = 0. Pìsa m mhdenik� dianÔsmata prokÔptoun apì th diadikasÐa

Gram-Schmidt?

'Askhsh 3.58 BreÐte orjog¸nia dianÔsmata w1, w2, w3 apì ta dianÔsmata

v1 = (1, −1, 0, 0), v2 = (0, 1, −1, 0), v3 = (0, 0, 1, −1) .

Ta v1, v2, v3 apoteloÔn b�sh tou upoq¸rou pou eÐnai orjog¸nioc sto (1, 1, 1, 1).


Upìloipa apì prohgoÔmenec dialèxeic

H akìloujh Prìtash dÐdei tic basikèc idiìthtec tou orjogwnÐou sumplhr¸matoc.

Prìtash 3.11 'Estw ènac dianusmatikìc upìqwroc V tou Rn, kai W to orjog¸nio su-

mpl rwma tou V , W = V ⊥. Tìte

1. H di�stash tou W eÐnai dimW = n− dimV , kai V ∩W = {0}.

2. To orjog¸nio sumpl rwma tou W eÐnai o V : e�n W = V ⊥ tìte V = W⊥.

3. E�n {v1, . . . , vk} eÐnai b�sh tou V kai {w1, . . . , wn−k} b�sh touW , tìte {v1, . . . , vk,w1, . . . , wn−k} eÐnai b�sh tou Rn.

4. K�je di�nusma x ∈ Rn, gr�fetai me monadikì trìpo wc �jroisma enìc dianÔsmatoc touV kai enìc dianÔsmatoc tou W .

Apìdeixh. 1. JewroÔme mia b�sh v1, . . . , vk tou V kai ton pÐnaka A pou èqei wc grammèc

ta dianÔsmata v1, . . . , vk. Tìte V eÐnai o q¸roc gramm¸n tou A, kai o mhdenoq¸roc tou A

eÐnai Ðsoc me to orjog¸nio sumpl rwma tou V , N (A) = W . 'Ara dimW = dimN (A) =

n− k.'Estw t¸ra di�nusma x ∈ V ∩W, x = (x1, x2, . . . , xn). AfoÔ x ∈ V kai x ∈ W = V ⊥,

to x eÐnai orjog¸nio ston eautì tou, xxT = 0. Dhlad x21 + x22 + · · ·x2n = 0 kai sunep¸c

x = 0.

2. Apì to Je¸rhma 3.4, to orjog¸nio sumpl rwma tou W = N (A) eÐnai o R(AT ) = V .

3. To sÔnolo {v1, . . . , vk, w1, . . . , wn−k} èqei n stoiqeÐa. E�n deÐxoume oti eÐnai grammik�

anex�rthta, tìte ja eÐnai b�sh tou Rn. Upojètoume oti a1v1 + · · · + akuk + b1w1 + · · · +bn−kwn−k) = 0. Jèloume na deÐxoume oti tìte ìla ta ai kai bj eÐnai 0. 'Eqoume

a1v1 + · · ·+ akuk = y = −(b1w1 + · · ·+ bn−kwn−k) .

All� h arister pleur� an kei sto V , h dexi� pleur� an kei sto W . 'Ara to di�nusma y

an kei sthn tom V ∩W , kai sunep¸c y eÐnai to mhdenikì di�nusma 0 ∈ Rn. 'Ara a1u1 +

· · ·+ akuk = 0, kai afoÔ u1, . . . , uk eÐnai grammik� anex�rthta, ìla ta ai eÐnai 0. Parìmoia,

b1w1 + · · ·+ bn−kwn−k = 0, kai afoÔ to w1, . . . , wn−k eÐnai grammik� anex�rthta, ìla ta bj

eÐnai 0.

4. AfoÔ {u1, . . . , uk, w1, . . . , wn−k} eÐnai b�sh tou Rn, k�je di�nusma x ∈ Rn gr�fetai

wc grammikìc sunduasmìc

x = a1u1 + · · ·+ akuk + b1w1 + · · ·+ bn−kwn−k

= x′ + x′′

ìpou x′ = a1u1 + · · ·+ akuk ∈ V kai x′′ = b1w1 + · · ·+ bn−kwn−k ∈W .

Upojètoume oti isqÔei epÐshc x = x+ x, ìpou x ∈ V kai x ∈W . Tìte x′ + x′′ = x+ x,

kai sunep¸c x′ − x = x− x′′, all� h arister pleur� an kei sto V , h dexi� pleur� an kei

sto W , kai ìpwc pio p�nw, eÐnai kai oi dÔo mhdèn. 'Ara x′ = x kai x′′ = x.

�


Di�lexh 17

Nìrma kai eswterikì ginìmeno se genikoÔc dianusmati-

koÔc q¸rouc

Sth sunèqeia ja orÐsoume eswterikì ginìmeno se genikoÔc dianusmatikoÔc q¸rouc, kai ja

melet soume k�poiec idiìthtèc tou sthn perÐptwsh pou to s¸ma K eÐnai oi pragmatikoÐ oi

migadikoÐ arijmoÐ:

Se autì to kef�laio, K = R C .

Gia to Rn, èqoume dei to eswterikì ginìmeno dÔo dianusm�twn x kai y,

x · y = xT y = x1y1 + · · ·+ xnyn

kai th nìrma

||x|| =√〈x, y〉 .

Sto C, ja jèlame h nìrma ||z|| na sumpÐptei me to mètro |z| kai, e�n z = x+ iy me th nìrma

tou (x, y):

||z|| = |z| =√zz =

√x2 + y2 = ||(x, y)|| .

An�loga, sto Cn, e�n jèloume ||(z1, . . . , zn)|| na sumpÐptei me th nìrma tou dianÔsmatoc

(x1, y1, x2, y2, . . . , xn, yn) sto R2n, ìpou zj = xj + iyj , prèpei na orÐsoume th nìrma

||z|| =√z1z1 + · · ·+ znzn

kai to eswterikì ginìmeno

〈z, w〉 = z1w1 + · · ·+ znwn .

Met� apì autèc tic parathr seic, dÐdoume ton akìloujo orismì.

Orismìc 3.4. V dianusmatikìc q¸roc p�nw apì to s¸ma K(= R C). Mia apeikìnish

V → R : v 7→ ||v|| onom�zetai nìrma ( st�jmh) e�n

N 1. ||v|| = 0 e�n kai mìnon e�n v = 0

N 2. Gia k�je v ∈ V kai a ∈ K, ||av|| = |a| ||v||

N 3. Gia k�je v, w ∈ V , ||v + w|| ≤ ||v||+ ||w|| (trigwnik anisìthta)

L mma 3.12 Se èna dianusmatikì q¸ro V me nìrma,

1. Gia k�je v ∈ V , ||v|| ≥ 0.

2. Gia k�je v, w ∈ V , ||v − w|| ≥ | ||v|| − ||w|| |

Apìdeixh.


1. Gia k�je v ∈ V ,

||v|| =1

2(||v||+ ||v||) =

1

2(||v||+ || − v||)

≥ 1

2||v + (−v)|| = 1

2||0|| = 0 .

2. Gia k�je v, w ∈ V ,

||v|| = ||(v − w) + w|| ≤ ||v − w||+ ||w|| ,

kai sunep¸c

||v − w|| ≥ ||v|| − ||w|| .

An�loga

||v − w|| = ||w − v|| ≥ ||w|| − ||v|| .

�

Par�deigma 3.10 Sto Rn kai sto Cn h eukleÐdeia nìrma ( `2-nìrma) eÐnai h sun jhc

nìrma

||x|| =√x21 + · · ·+ x2n

kai

||z|| =√z1z1 + · · ·+ znzn .

Par�deigma 3.11 H `1-nìrma sto Rn orÐzetai wc

||x||1 = |x1|+ · · ·+ |xn| .

Elègqoume ta axi¸mata:

N 1.

||x||1 = 0 ⇔ |x1| = · · · = |xn| = 0

⇔ x = 0 .

N 2.

||ax||1 =

n∑i=1

|axi| = |a|n∑i=1

|xi| = |a| ||x||1 .

N 3.

||x+ y||1 =n∑i=1

|xi + yi| ≤n∑i=1

|xi|+ |yi| = ||x||1 + ||y||1 .


Par�deigma 3.12 H `∞-nìrma sto Rn orÐzetai wc

||x||∞ = max{|xi| : i = 1, . . . , n} .

'Askhsh 3.59 DeÐxte oti ||x||∞ ikanopoieÐ ta axi¸mata thc nìrmac.

Par�deigma 3.13 Sto q¸ro twn poluwnÔmwn K[x], me K = R C, orÐzoume th nìrma

||p(x)|| =(∫ 1

0|p(t)|2dt

)1/2

.

O èlegqoc twn axiwm�twn N 1 kai N 2 eÐnai eÔkoloc. Gia to N 3 parathroÔme oti

||p(x) + q(x)||2 =

∫ 1

0|p(t) + q(t)|2dt

=

∫ 1

0|p(t)|2dt+

∫ 1

0|q(t)|2dt+ 2Re

∫ 1

0p(t)q(t)dt

en¸

(||p(x)||+ ||q(x)||)2 = ||p(x)||2 + ||q(x)||2 + 2||p(x)|| ||q(x)|| .

'Ara gia na isqÔei h trigwnik anisìthta, arkeÐ na isqÔei h anisìthta

Re

∫ 1

0p(t)q(t)dt ≤ ||p(x)|| ||q(x)|| ,

thn opoÐa ja apodeÐxoume sthn Prìtash 3.14.

Par�deigma 3.14 Sto q¸ro C[a, b] twn suneq¸n sunart sewn sto di�sthma [a, b] (me

pragmatikèc migadikèc timèc) orÐzoume thn L2-nìrma

||f || =(∫ b

a|f(t)|2dt

)1/2

kai thn L∞-nìrma

||f ||∞ = max{|f(x)| : x ∈ [a, b]} .

'Askhsh 3.60 DeÐxte oti ||f ||∞ ikanopoieÐ ta axi¸mata thc nìrmac.

Eswterikì ginìmeno

Sth sunèqeia ja qrhsimopoioÔme to sumbolismì tou suzugoÔc, a, katano¸ntac oti e�n to

s¸ma K eÐnai oi pragmatikoÐ arijmoÐ, tìte a = a.


Orismìc 3.5. JewroÔme dianusmatikì q¸ro V p�nw apì to s¸ma K (= R C). Mia

apeikìnish

V × V → K : (v, w) 7→ 〈v, w〉

onom�zetai eswterikì ginìmeno e�n

EG 1. EÐnai grammik sthn pr¸th metablht , dhlad e�n gia k�je u, v, w ∈ V kai a ∈ K,

〈u+ v, w〉 = 〈u, w〉+ 〈v, w〉

kai

〈av, w〉 = a〈v, w〉 .

EG 2. Gia k�je v, w ∈ V ,〈v, w〉 = 〈w, v〉

EG 3. Gia k�je v ∈ V , e�n v 6= 0, tìte 〈v, v〉 > 0.

E�n 〈v, w〉 = 0, lème oti ta dianÔsmata v kai w eÐnai orjog¸nia.

ParathroÔme oti e�n to s¸ma K = R, tìte h idiìthta EG 2 shmaÐnei oti to eswterikì

ginìmeno eÐnai summetrikì, kai mazÐ me thn EG 1, oti eÐnai grammikì kai sth deÔterh metablht .

Antijètwc, e�n K = C, gia th deÔterh metablht èqoume

〈v, aw〉 = 〈aw, v〉

= a〈w, v〉

= a〈v, w〉 .

Par�deigma 3.15 Sto Rn orÐzetai to eukleÐdeio eswterikì ginìmeno: e�n x = (x1, . . . , xn), y =

(y1, . . . , yn) ,

〈x, y〉 =n∑i=1

xiyi .

Ta dianÔsmata (x1, x2) kai (−x2, x1) eÐnai orjog¸nia dianÔsmata sto R2 me to eukleÐdeio

eswterikì ginìmeno.

Par�deigma 3.16 Sto Cn orÐzetai eswterikì ginìmeno gia z = (z1, . . . , zn), w =

(w1, . . . , wn) ,

〈z, w〉 =n∑i=1

ziwi .

Drasthriìthta 3.6 Ta dianÔsmata (z1, z2) kai (−z2, z1) den eÐnai orjog¸nia

sto C2 me autì to eswterikì ginìmeno. BreÐte èna mh mhdenikì di�nusma orjog¸nio

sto (z1, z2)


Par�deigma 3.17 JewroÔme pÐnaka A =

[a b

c d

], kai orÐzoume thn apeikìnish f :

K2 ×K2 −→ K,

f(v, w) = vTAw (3.4)

= [v1, v2]

[a b

c d

][w1

w2

](3.5)

= av1w1 + bv1w2 + cv2w1 + dv2w2 . (3.6)

Elègxte oti ikanopoieÐtai h idiìthta EG1. Gia thn EG2, aw1v1 + bw1v2 + cw2v1 + dw2v2 =

av1w1 + bv1w2 + cv2w1 + dv2w2prèpei na apait soume a = a, d = d kai c = b. Tèloc, gia na

isqÔei h EG3, prèpei na isqÔei av1v1 + bv1v2 + cv2v1 + dv2v2 > 0 gia k�je v ∈ K2, v 6= 0.

JewroÔme v = (1, 0) v = (0, 1), kai èqoume a > 0, d > 0. MporoÔme na deÐxoume oti f

ikanopoieÐ thn idiìthta EG3 e�n kai mìnon e�n ad − bb > 0. (Dec thn 'Askhsh 3.70 gia thn

pragmatik perÐptwsh.)

Par�deigma 3.18 Sto q¸ro twn poluwnÔmwn K[x] orÐzoume to eswterikì ginìmeno

〈p(x), q(x)〉 =

∫ 1

0p(t) q(t) dt (3.7)

Drasthriìthta 3.7 Elègxte oti h 3.7 pr�gmati orÐzei èna eswterikì ginìmeno.

Ta polu¸numa p(x) = (x− 12)2 kai q(x) = (x− 1

2)3 eÐnai orjog¸nia:

〈p(x), q(x)〉 =

∫ 1

0(t− 1

2)2(t− 12)3dt

=

∫ 1

0(t− 1

2)5dt

= 0 .

Par�deigma 3.19 Sto q¸ro twn suneq¸n sunart sewn sto di�sthma [a, b], me prag-

matikèc timèc, C[a, b], me migadikèc timèc , CC[a, b], orÐzoume to eswterikì ginìmeno

〈f, g〉 =

∫ b

af(s) g(s) ds (3.8)

Drasthriìthta 3.8 Elègxte oti h 3.8 pr�gmati orÐzei èna eswterikì ginìmeno.

Oi sunart seic sin kai cos eÐnai orjog¸niec sto C[0, π]:∫ π

0sin t cos tdt =

1

2

∫ π

0sin 2tdt = 0 .


Je¸rhma 3.13 (Anisìthta Cauchy-Schwarz) Se èna q¸ro me eswterikì ginìmeno

isqÔei, gia k�je v, w:

|〈v, w〉| ≤√〈v, v〉

√〈w, w〉 .

Apìdeixh. E�n w = 0, tìte kai oi dÔo pleurèc mhdenÐzontai kai h sqèsh epalhjeÔetai.

Upojètoume oti w 6= 0 kai jewroÔme, gia a ∈ K, to di�nusma v − aw:

0 ≤ 〈v − aw, v − aw〉

= 〈v, v〉 − 〈v, aw〉 − 〈aw, v〉+ 〈aw, aw〉

= 〈v, v〉 − a〈v, w〉 − a〈v, w〉+ aa〈w, w〉 .

Eidikìtera, gia a = 〈v, w〉〈w,w〉 èqoume

〈v, v〉 ≥ 〈v, w〉〈v, w〉〈w, w〉

|〈v, w〉|2 ≤ 〈v, v〉〈w, w〉 ,

kai afoÔ oi pragmatikoÐ arijmoÐ 〈v, v〉, 〈w, w〉 kai |〈v, w〉| eÐnai jetikoÐ mhdèn, èqoume

|〈v, w〉| ≤√〈v, v〉

√〈w, w〉 .

�

Eidikèc peript¸seic thc anisìthtac Cauchy-Schwarz eÐnai oi akìloujec anisìthtec.

Sto q¸ro Rn Cn, me to sun jec eswterikì ginìmeno 〈x, y〉 =∑n

i=1 xiyi,∣∣∣∣∣n∑i=1

xiyi

∣∣∣∣∣ ≤(

n∑i=1

|xi|2)1/2( n∑

i=1

|yi|2)1/2

.

Sto q¸ro C[a, b], me eswterikì ginìmeno 〈f, g〉 =∫ ba f(s)g(s)ds ,∣∣∣∣∫ b

af(s)g(s)ds

∣∣∣∣ ≤ (∫ b

a(f(s))2 ds

)1/2(∫ b

a(g(s))2 ds

)1/2

.

Prìtash 3.14 E�n V eÐnai q¸roc me eswterikì ginìmeno, tìte orÐzetai mÐa nìrma sto

V :

||v|| =√〈v, v〉 .

Apìdeixh. H apìdeixh twn N 1 kai N 2 eÐnai apl . Gia na apodeÐxoume thn trigwnik

anisìthta N 3, parathroÔme oti

||v + w||2 = 〈v + w, v + w〉

= ||v||2 + ||w||2 + 2Re 〈v, w〉 ,


kai oti

(||v||+ ||w||)2 = ||v||2 + ||w||2 + 2||v|| ||w|| .

All� apì thn anisìthta Cauchy-Schwarz, 〈v, w〉 ≤ ||v|| ||w|| kai sunep¸c

||v + w||2 ≤ (||v||+ ||w||)2 .

AfoÔ oi pragmatikoÐ arijmoÐ ||v + w||, ||v|| kai ||w|| eÐnai jetikoÐ mhdèn, èqoume

||v + w|| ≤ ||v||+ ||w|| .

�

Me ton sumbolismì thc nìrmac, h anisìthta Cauchy-Schwarz gr�fetai sth morf

|〈v, w〉| ≤ ||v|| ||w|| .

Jewr¸ntac th nìrma wc to m koc tou dianÔsmatoc, mporoÔme na orÐsoume th gwnÐa metaxÔ

dÔo dianusm�twn se opoiod pote pragmatikì dianusmatikì q¸ro me eswterikì ginìmeno.

Apì thn anisìthta Cauchy-Schwarz èqoume

−1 ≤ 〈v, w〉||v|| ||w||

≤ 1 ,

kai sunep¸c up�rqei ϑ ∈ [0, π] tètoio ¸ste

cosϑ =〈v, w〉||v|| ||w||

. (3.9)

Autìc o orismìc gwnÐac tairi�zei me thn ènnoia thc orjogwniìthtac, pou èqoume orÐsei se

opoiod pote q¸ro me eswterikì ginìmeno, pragmatikì migadikì. 'Otan 〈v, w〉 = 0, ϑ = π2

kai ta dianÔsmata v kai w sqhmatÐzoun orj gwnÐa.

Sthn eukleÐdeia gewmetrÐa, o �nìmoc tou parallhlogr�mmou� lèei oti se èna parallhlì-

grammo, to �jroisma twn tetrag¸nwn twn tess�rwn pleur¸n isoÔtai me to �jroisma twn

tetrag¸nwn twn diagwnÐwn. 'Otan to parallhlìgrammo eÐnai orjog¸nio, autì eÐnai isodÔ-

namo me to Pujagìreio Je¸rhma. Ja doÔme oti èna an�logo apotèlesma isqÔei se k�je

dianusmatikì q¸ro me eswterikì ginìmeno.

Prìtash 3.15 'Estw V ènac q¸roc me eswterikì ginìmeno.

1. Nìmoc tou Parallhlogr�mmou. Gia k�je v, w ∈ V isqÔei h isìthta

||v + w||2 + ||v − w||2 = 2||v||2 + 2||w||2 .

2. Pujagìreio Je¸rhma. E�n v1, . . . , vk ∈ V kai 〈vi, vj〉 = 0 ìtan i 6= j, tìte

||v1 + · · ·+ vk||2 = ||v1||2 + · · ·+ ||vk||2 .


Apìdeixh. Se èna q¸ro me eswterikì ginìmeno èqoume ||v||2 = 〈v, v〉. Sunep¸c

||v + w||2 = 〈v + w, v + w〉 = 〈v, v〉+ 〈v, w〉+ 〈w, v〉+ 〈w, w〉 , (3.10)

||v − w||2 = 〈v − w, v − w〉 = 〈v, v〉 − 〈v, w〉 − 〈w, v〉+ 〈w, w〉 . (3.11)

Prosjètontac tic 3.10 kai 3.11 èqoume ton �nìmo tou parallhlogr�mmou�.

||v + w||2 + ||v − w||2 = 2〈v, v〉+ 2〈w, w〉 .

Gia k = 2 to �Pujagìreio Je¸rhma� prokÔptei apì thn 3.10, afoÔ 〈v1, v2〉 = 0. H genik

perÐptwsh apodeiknÔetai me epagwg sto k. AfoÔ 〈v1 + · · ·+ vk−1, vk〉 = 0,

||v1 + · · ·+ vk||2 = ||v1 + · · ·+ vk−1||2 + ||vk||2

= ||v1||2 + · · ·+ ||vk−1||2 + ||vk||2 .

�

Par�deigma 3.20 Ja deÐxoume oti h `1 nìrma sto R2 den ikanopoieÐ to nìmo tou pa-

rallhlogr�mmou. JewroÔme ta dianÔsmata v = (1, 0) kai w = (0, 1). Tìte ||v||1 =

|1| + |0| = 1, ||w||1 = 1, kai ||v + w||1 = 2, ||v − w||1 = 2. 'Ara 2||v||2 + 2||w||2 = 4

en¸ ||v + w||2 + ||v − w||2 = 8.

AfoÔ k�je nìrma pou proèrqetai apì eswterikì ginìmeno ikanopoieÐ to Nìmo tou Paral-

lhlogr�mmou, sumperaÐnoume oti h nìrma `1 den prokÔptei apì eswterikì ginìmeno.


Di�lexh 18

Orjokanonik� sÔnola dianusm�twn

'Ena sÔnolo dianusm�twn S = {v1, . . . , vn} onom�zetai orjog¸nio e�n ta stoiqeÐa tou

eÐnai orjog¸nia an� dÔo, dhlad e�n gia k�je i, j = 1, . . . , n, i 6= j ,

〈vi, vj〉 = 0 .

E�n epÐ plèon, gia k�je i = 1, . . . , n, ||vi|| = 1, to sÔnolo onom�zetai orjokanonikì.

Qrhsimopoi¸ntac to sumbolismì δ tou Kronecker,

δij =

[0 i 6= j

1 i = j

blèpoume oti to sÔnolo S eÐnai orjokanonikì e�n kai mìnon e�n, gia k�je i, j = 1, . . . , n ,

〈vi, vj〉 = δij .

L mma 3.16 'Ena orjokanonikì sÔnolo eÐnai grammik� anex�rthto.

Apìdeixh. Upojètoume oti {v1, . . . , vn} eÐnai èna orjokanonikì sÔnolo, kai oi arijmoÐ

a1, . . . , an eÐnai tètoioi ¸ste

a1v1 + · · ·+ anvn = 0 .

Gia k�je j = 1, . . . , n, èqoume

0 = 〈0, vj〉

=

⟨n∑i=1

aivi, vj

⟩

=

n∑i=1

ai〈vi, vj〉

=

n∑i=1

aiδij

= aj .

Sunep¸c aj = 0 gia k�je j = 1, . . . , n, kai to sÔnolo {v1, . . . , vn} eÐnai grammik� anex�r-

thto.

�

'Ena orjokanonikì sÔnolo apoteleÐ mÐa idiaÐtera qr simh b�sh gia to q¸ro ton opoÐo

par�gei. Oi suntetagmènec enìc dianÔsmatoc wc proc mÐa orjokanonik b�sh dÐdontai apl¸c

apì ta eswterik� ginìmena tou dianÔsmatoc me ta dianÔsmata thc b�shc. Pr�gmati, e�n

{v1, . . . , vn} eÐnai orjokanonik b�sh, kai

v = a1v1 + · · ·+ anvn


tìte

〈v, vj〉 =

⟨n∑i=1

aivi , vj

⟩

=n∑i=1

ai〈vi, vj〉

=n∑i=1

aiδij

= aj .

Se èna q¸ro peperasmènhc di�stashc me eswterikì ginìmeno, mporoÔme p�nta na kata-

skeu�soume mia orjokanonik b�sh, efarmìzontac th diadikasÐa orjokanonikopoÐhshc

Gram-Schmidt.

Je¸rhma 3.17 (OrjokanonikopoÐhsh Gram-Schmidt) JewroÔme q¸ro V me e-

swterikì ginìmeno, kai èna grammik� anex�rthto sÔnolo {v1, . . . , vn}. Tìte up�rqei orjo-kanonikì sÔnolo {e1, . . . , en} tètoio ¸ste gia k�je i = 1, . . . , n

ei ∈ 〈v1, . . . , vi〉

kai

〈e1, . . . , en〉 = 〈v1, . . . , vn〉 .

Sthn apìdeixh tou Jewr matoc o sumbolismìc 〈. . .〉 qrhsimopoieÐtai gia na dhl¸sei tìso

to eswterikì ginìmeno ìso kai ton paragìmeno upìqwro, all� h di�krish eÐnai sun jwc

eÔkolh apì ta sumfrazìmena.

Apìdeixh. Gia k�je j orÐzoume pr¸ta to di�nusma e′j orjog¸nio proc ta ei gia i =

1, . . . , j− 1, kai met� diairoÔme me th nìrma tou e′j gia na p�roume to monadiaÐo di�nusma ej .

èna sÔnolo mh mhdenik¸n orjogwnÐwn dianusm�twn e′1, . . . , e′n, kai sth sunèqeia ja orÐsoume

ta monadiaÐa dianÔsmata

ei =1

||e′i||e′i .

AfoÔ ta {v1, . . . , vn} eÐnai grammik� anex�rthta, v1 6= 0, kai orÐzoume

e′1 = v1 , e1 =1

||e′1||e′1 .

To e′2 prokÔptei apì to v2 afair¸ntac kat�llhlo pollapl�sio tou e1 ¸ste e′2 na eÐnai

orjog¸nio proc to e1.

e′2 = v2 − 〈v2, e1〉e1 .

Pr�gmati

〈e′2, e′1〉 = 〈v2 − 〈v2, e1〉 e1, e1〉

= 〈v2, e1〉 − 〈v2, e1〉〈e1, e1〉

= 0


ParathroÔme oti afoÔ ta e1, v2 eÐnai grammik� anex�rthta, e′2 6= 0 kai mporoÔme na diairè-

soume to e′2 me th nìrma tou gia na p�roume to monadiaÐo

e2 =1

||e′2||e′2 .

ParathroÔme oti ta e1, e2 par�goun ton Ðdio upìqwro pou par�goun ta v1, v2.

Sth sunèqeia, gia j = 3, . . . , n, orÐzoume anadromik� ta mh mhdenik� dianÔsmata

e′j = vj −j−1∑i=1

〈vj , ei〉 ei , (3.12)

ta opoÐa ikanopoioÔn 〈ei, e′j〉 = 0 gia i = 1, . . . , j−1. To e′j den eÐnai mhdenikì, �ra mporoÔme

na to diairèsoume me th nìrma tou gia na p�roume to di�nusma

ej =1

||e′j ||e′j .

Apì thn 3.12 eÐnai fanerì oti

ej ∈ 〈v1, . . . , vj〉

kai oti

vj ∈ 〈e1, . . . , ej〉 .

Sunep¸c, gia k�je j = 1, . . . , n

〈e1, . . . , ej〉 = 〈v1, . . . , vj〉 .

�

Parat rhsh 'Otan k�noume upologismoÔc me to qèri, eÐnai suqn� protimìtero na upo-

logÐsoume ìla ta e′j , akolouj¸ntac th diadikasÐa Gram – Schmidt ìpwc thn perigr�yame

sto Par�deigma 3.9, kai sto tèloc na diarèsoume k�je di�nusma e′i me th nìrma tou. Me

autìn ton trìpo apofeÔgoume na metafèroume se ìlh th diadikasÐa tic tetragwnikèc rÐzec

pou prokÔptoun apì th nìrma.

Polu¸numa Legendre

H sun jhc b�sh tou q¸rou twn poluwnÔmwn, {1, x, x2, . . .}, den eÐnai orjog¸nia gia opoio-

d pote eswterikì ginìmeno thc morf c 〈p, q〉 =∫ ba p(t)q(t)dt.

Gia na orÐsoume mÐa orjokanonik oikogèneia poluwnÔmwn, pou ja mac epitrèpei na pro-

seggÐzoume sunart seic apì tic probolèc touc sta stoiqeÐa aut c thc oikogèneiac, dieu-

kolÔnei na qrhsimopoi soume èna eswterikì ginìmeno pou orÐzetai sto di�sthma [−1, 1].

Me aut thn epilog , k�je �rtio polu¸numo, gia to opoÐo p(x) = p(−x), eÐnai ex arq c

orjog¸nio se k�je perittì polu¸numo, gia to opoÐo p(x) = −p(−x).


JewroÔme loipìn to eswterikì ginìmeno sto q¸ro R[x],

〈p, q〉 =

∫ 1

−1p(t)q(t)dt ,

kai ta polu¸numa p0(x) = 1, p1(x) = x, p2(x) = x2, . . . , pn(x) = xn, . . . .

OrÐzoume

w0(x) = p0(x) = 1 .

'Eqoume 〈p1, w0〉 =∫ 1−1 t dt = 0. 'Ara

w1(x) = p1(x)− 〈p1, w0〉〈w0, w0〉

w0(x) = x .

Gia na broÔme to w2, upologÐzoume ta eswterik� ginìmena

〈p2, w1〉 =

∫ 1

−1t3 dt = 0 ,

〈p2, w0〉 =

∫ 1

−1t2 dt =

1

3t3∣∣∣∣1−1

=2

3,

〈w0, w0〉 =

∫ 1

−11 dt = 2 ,

〈w1, w1〉 =

∫ 1

−1t2 dt =

2

3.

'Etsi èqoume

w2(x) = p2(x)− 〈p2, w1〉〈w1, w1〉

w1(x)− 〈p2, w0〉〈w0, w0〉

w0(x)

= x2 − 2/3

2= x2 − 1

3.

kai

〈w2, w2〉 =

∫ 1

−1

(t2 − 1

3

)2

dt =8

45.

'Otan diairèsoume ta w0, w1, w2 me th nìrma touc, èqoume

q0(x) =1√2,

q1(x) =

√3√2x ,

q2(x) =3√

5√8

(x2 − 1

3

)=

√5√8

(3x2 − 1) .

Aut� sundèontai me ta polu¸numa Legendre,

Pm(x) =1

2mm!

dm

dxm(x2 − 1)m .


Gia m = 0, . . . , n− 1, ta polu¸numa

qm(x) =

√2m+ 1

2Pm(x)

eÐnai h orjokanonik b�sh tou Rn−1[x] pou prokÔptei me th diadikasÐa Gram – Schmidt, gia

to eswterikì ginìmeno sto di�sthma [−1, 1], apì th sun jh b�sh {1, x, x2, . . . , xn−1}.

'Askhsh 3.61 JewroÔme dianÔsmata v = (v1, v2) kai w = (w1, w2) sto R2.

1. DeÐxte oti h sun�rthsh

(v, w) = 4v1w1 + 9v2w2

orÐzei eswterikì ginìmeno sto R2.

2. DeÐxte oti h sun�rthsh

(v, w) = 2v1w1 − v2w2

den orÐzei eswterikì ginìmeno.

'Askhsh 3.62 JewroÔme to q¸ro C[0, 1] twn suneq¸n sunart sewn sto di�sthma

[0, 1], me eswterikì ginìmeno

(f, g) =

∫ 1

0f(t) g(t) dt

1. BreÐte to eswterikì ginìmeno twn f(x) = 2x+ 1, g(x) = 3x− 2.

2. DeÐxte oti oi sunart seic f(x) = x2 kai g(x) = 4x− 3 eÐnai orjog¸niec.

3. BreÐte mÐa sun�rthsh orjog¸nia proc thn f(x) = 6x+ 12

'Askhsh 3.63 JewroÔme to migadikì dianusmatikì q¸ro C2, me to sÔnhjec esw-

terikì ginìmeno. BreÐte ta (u, v), ||u||, ||v|| kai thn apìstash d(u, v) = ||u− v|| giata dianÔsmata:

1. u = (2− i, 3 + 2i), v = (3− 2i, 2 + i).

2. u = (2− 3i, −2 + 3i), v = (1, 1).

'Askhsh 3.64 Sto q¸ro R3, me to eukleÐdeio eswterikì ginìmeno, breÐte thn

orjokanonik b�sh pou prokÔptei apì thn efarmog thc diadikasÐac Gram-Schmidt

sth b�sh

{(1, 1, 0), (2, 1, 0), (0, 1, 2)} .


'Askhsh 3.65 Jewr ste jetik suneq sun�rthsh f : [0, 1] → R. DeÐxte oti

e�n p(x), q(x) eÐnai polu¸numa, tìte

〈p, q〉f =

∫ 1

0f(t)p(t)q(t)dt ,

orÐzei eswterikì ginìmeno sto q¸ro twn poluwnÔmwn me pragmatikoÔc suntelestèc.

E�n f(x) = x + 1, breÐte orjokanonik b�sh gia to q¸ro twn poluwnÔmwn bajmoÔ

to polÔ 1, me to eswterikì ginìmeno 〈 , 〉f .

'Askhsh 3.66 BreÐte ìla ta diaforetik� eswterik� ginìmena pou orÐzontai se èna

dianusmatikì q¸ro di�stashc 2 p�nw apì to R.

'Askhsh 3.67 JewroÔme dianusmatikì q¸ro V p�nw apì to R, me eswterikì

ginìmeno, kai dÔo diaforetik� dianÔsmata a, b ∈ V . ApodeÐxte oti e�n x ∈ V kai

||x− a||+ ||x− b|| = ||a− b||, tìte x = λa+ µb, me λ, µ ∈ R kai λ+ µ = 1.

'Askhsh 3.68 JewroÔme ton dianusmatikì q¸ro R3, me to kanonikì eswterikì

ginìmeno.

1. EpalhjeÔste oti ta dianÔsmata

v1 = (1, 1, 1), v2 = (1, 2, −3), v3 = (5, −4, −1)

eÐnai an� dÔo orjog¸nia, kai breÐte mÐa orjokanonik b�sh tou V , diaforetik

apì thn kanonik .

2. BreÐte ta monadiaÐa dianÔsmata ta opoÐa eÐnai tautìqrona orjog¸nia me ta

v1 − v2 kai v1 + v3.

3. BreÐte ta dianÔsmata ta opoÐa eÐnai orjog¸nia sto 2v2 + v3 kai an koun ston

grammikì upìqwro pou par�getai apì ta v1 − v2, v1 + v3.

'Askhsh 3.69 JewroÔme ton dianusmatikì q¸ro R4 me to kanonikì eswterikì

ginìmeno, kai ton upìqwro X pou par�getai apì ta dianÔsmata u1 = (1, 1, 0, 0) kai

u2 = (0, 1, −1, 1).

BreÐte mÐa orjokanonik b�sh tou orjogwnÐou sumplhr¸matoc X⊥, kai sumplhr¸ste

thn se mia orjokanonik b�sh tou R4.


'Askhsh 3.70 'Estw 2× 2 pragmatikìc pÐnakac A. DeÐxte oti h apeikìnish

〈x, y〉 = xTAy = [x1, x2]

[a b

c d

][y1

y2

]

orÐzei èna eswterikì ginìmeno sto R2 e�n kai mìnon e�n AT = A, detA > 0 kai

trA > 0, (dhlad e�n b = c, ad− b2 > 0 kai a > 0).

Kef�laio 4

Telestèc se q¸ro me eswterikì

ginìmeno

126

Kef�laio 4 Telestèc se q¸ro me eswterikì ginìmeno 127

Di�lexh 19

ErmitianoÐ telestèc

Gia ènan telest L se èna dianusmatikì q¸ro peperasmènhc di�stashc, èqoume dei oti e�n

up�rqei b�sh tou V h opoÐa apoteleÐtai apì idiodianÔsmata tou L tìte o pÐnakac tou L wc

proc aut th b�sh eÐnai diag¸nioc. T¸ra ja doÔme oti se q¸rouc me eswterikì ginìmeno

mporoÔme na d¸soume sugkekrimèna krit ria gia na sumbaÐnei autì.

Se autì to kef�laio ìloi oi dianusmatikoÐ q¸roi eÐnai p�nw apì to s¸ma C to s¸ma R.

Orismìc 4.1. JewroÔme èna dianusmatikì q¸ro V me eswterikì ginìmeno kai èna

grammikì telest L : V → V . O telest c L onom�zetai ermitianìc e�n gia k�je

u, v ∈ V ,〈L(u), v〉 = 〈u, L(v)〉 .

'Enac ermitianìc telest c se èna pragmatikì dianusmatikì q¸ro onom�zetai summetri-

kìc.

Par�deigma 4.1 JewroÔme ton telest L : R2 → R2, L(x, y) = (x+ 2y, 2x). Wc proc

to eukleÐdeio eswterikì ginìmeno èqoume

〈L(u1, u2), (v1, v2)〉 = 〈(u1 + 2u2, 2u1), (v1, v2)〉 = u1v1 + 2u2v1 + 2u1v2

kai

〈(u1, u2), L(v1, v2)〉 = 〈(u1, u2), (v1 + 2v2, 2v1)〉 = u1v1 + 2u1v2 + 2u2v1 .

O telest c L eÐnai summetrikìc.

Par�deigma 4.2 JewroÔme ton telest M : C2 → C2, L(z, w) = (z + iw, −iz). Wc

proc to sunhjismèno eswterikì ginìmeno sto C2 èqoume

〈L(u1, u2), (v1, v2)〉 = 〈(u1 + iu2,−iu1), (v1, v2)〉 = u1v1 + iu2v1iu1v2

kai

〈(u1, u2), L(v1, v2)〉 = 〈(u1, u2), (v1 + iv2, −iv1)〉 = u1v1 + u1(−i)v2 − u2iv1 .

Orismìc 4.2. JewroÔme ènan tetragwnikì migadikì pÐnaka A = [aij ]. O suzug c (

anastrofosuzug c) tou pÐnaka A eÐnai o pÐnakac A∗ = [bij ], ìpou

bij = aji .

Dhlad oi ìroi tou pÐnaka A∗ eÐnai oi migadikoÐ suzugeÐc twn ìrwn tou an�strofou tou A.

E�n o pÐnakac A eÐnai pragmatikìc, tìte A∗ = AT .


Par�deigma 4.3 'Estw o pÐnakac

A =

[1 i

2 3 + i

].

Tìte

AT =

[1 2

i 3 + i

], A =

[1 −i

2 3− i

]kai

A∗ = (A)T =

[1 2

−i 3− i

].

Orismìc 4.3. 'Enac tetragwnikìc migadikìc pÐnakac A onom�zetai ermitianìc e�n

eÐnai Ðsoc me ton suzug tou, A∗ = A.

'Enac ermitianìc pÐnakac tou opoÐou ìloi oi ìroi eÐnai pragmatikoÐ arijmoÐ eÐnai summetri-

kìc.

ParathroÔme oti ta diag¸nia stoiqeÐa enìc ermitianoÔ pÐnaka eÐnai pragmatikoÐ arijmoÐ.

L mma 4.1 JewroÔme ermitianì telest L : V → V se q¸ro peperasmènhc di�stashc,

kai orjokanonik b�sh B tou V . Tìte o pÐnakac A = [aij ] tou L wc proc th b�sh B eÐnai

ermitianìc,

aij = aji .

Apìdeixh. JewroÔme thn orjokanonik b�sh B = {u1, . . . , un}. E�n A = [aij ] eÐnai o

pÐnakac tou L wc proc th b�sh B, tìte gia k�je j = 1, . . . , n,

L(uj) =

n∑k=1

akjuk .

All� tìte

〈L(uj), ui〉 =

⟨n∑k=1

akjuk, ui

⟩

=

n∑k=1

〈akjuk, ui〉

=

n∑k=1

akj〈uk, ui〉

= aij ,

afoÔ h b�sh eÐnai orjokanonik . Ex �llou,

〈uj , L(ui)〉 =

⟨uj ,

n∑k=1

akiuk

⟩


=n∑k=1

〈uj , akiuk〉

=n∑k=1

aki〈uj , uk〉

= aji .

AfoÔ o L eÐnai ermitianìc, aij = 〈L(uj), ui〉 = 〈uj , L(ui)〉 = aji kai o pÐnakac A eÐnai Ðsoc

me ton suzug tou.

�

Prìtash 4.2 JewroÔme migadikì dianusmatikì q¸ro V kai telest L : V → V . E�n o

L eÐnai ermitianìc, tìte oi idiotimèc tou L eÐnai pragmatikoÐ arijmoÐ.

Apìdeixh. 'Estw λ ∈ C mÐa idiotim tou L, kai v ∈ V èna idiodi�nusma gia thn idiotim λ,

L(v) = λv. Tìte

〈L(v), v〉 = 〈λv, v〉 = λ〈v, v〉 ,

kai

〈v, L(v)〉 = 〈v, λv〉 = λ〈v, v〉 .

AfoÔ 〈v, v〉 6= 0 kai o L eÐnai ermitianìc, λ = λ. 'Ara h idiotim λ eÐnai pragmatikìc arijmìc.

�

L mma 4.3 E�n L : V → V eÐnai ermitianìc telest c, tìte ta idiodianÔsmata pou anti-

stoiqoÔn se diaforetikèc idiotimèc eÐnai orjog¸nia.

Apìdeixh. JewroÔme idiotimèc λ kai µ tou L kai antÐstoiqa idiodianÔsmata u kai v. Tìte

λ〈u, v〉 = 〈L(u), v〉 = 〈u, L(v)〉 = µ〈u, v〉 .

AfoÔ oi idiotimèc eÐnai pragmatikèc, λ〈u, v〉 = µ〈u, v〉, kai e�n λ 6= µ, 〈u, v〉 = 0.

�

MonadiaÐoi telestèc

Orismìc 4.4. JewroÔme èna dianusmatikì q¸ro V me eswterikì ginìmeno kai èna gram-

mikì telest L : V → V . O telest c L onom�zetai monadiaÐoc ( orjomonadiaÐoc)

e�n diathreÐ to eswterikì ginomeno, dhlad e�n gia k�je u, v ∈ V ,

〈L(u), L(v)〉 = 〈u, v〉 .

'Enac monadiaÐoc telest c se èna pragmatikì dianusmatikì q¸ro onom�zetai orjog¸nioc.


Par�deigma 4.4 O telest c L(x, y) = (x cosϑ − y sinϑ, x sinϑ + y cosϑ) eÐnai orjo-

g¸nioc. O telest c M(z, w) = (eiϑz, e−iϑw) eÐnai monadiaÐoc. Elègxte oti diathroÔn to

eswterikì ginìmeno sto R2 kai sto C2 antÐstoiqa.

Orismìc 4.5. 'Enac tetragwnikìc migadikìc pÐnakac A onom�zetai orjomonadiaÐoc

e�n A∗A = In.

'Enac orjomonadiaÐoc pÐnakac tou opoÐou ìloi oi ìroi eÐnai pragmatikoÐ arijmoÐ onom�zetai

orjog¸nioc.

Par�deigma 4.5 'Enac 2 × 2 pragmatikìc pÐnakac eÐnai orjog¸nioc e�n kai mìnon e�n

eÐnai thc morf c [cosϑ − sinϑ

sinϑ cosϑ

]

[cosϑ sinϑ

sinϑ − cosϑ

]

gia k�poio ϑ.

L mma 4.4 JewroÔme monadiaÐo telest L : V → V se q¸ro peperasmènhc di�stashc

n, kai orjokanonik b�sh B tou V . Tìte o pÐnakac A = [aij ] tou L wc proc th b�sh B eÐnai

orjomonadiaÐoc,n∑k=1

akiakj = δij .

Apìdeixh. JewroÔme thn orjokanonik b�sh B = {u1, . . . , un}. E�n A = [aij ] eÐnai o

pÐnakac tou L wc proc th b�sh B, tìte gia k�je j = 1, . . . , n,

L(uj) =

n∑k=1

akjuk .

AfoÔ o L eÐnai monadiaÐoc kai h b�sh B eÐnai orjokanonik , 〈L(ui), L(uj)〉 = 〈ui, uj〉 = δij .

Ex �llou

〈L(ui), L(uj)〉 =

⟨n∑k=1

akiuk,n∑`=1

a`ju`

⟩

=n∑k=1

n∑`=1

akia`j〈uk, u`〉

=

n∑k=1

n∑`=1

akia`jδk`

=n∑k=1

akiakj

= (A∗A)ji .


SumperaÐnoume oti A∗A = In kai o pÐnakac A eÐnai orjomonadiaÐoc.

�

Prìtash 4.5 JewroÔme migadikì dianusmatikì q¸ro V kai telest L : V → V . E�n o

L eÐnai monadiaÐoc, tìte oi idiotimèc tou L eÐnai migadikoÐ arijmoÐ mètrou 1.

Apìdeixh. 'Estw λ ∈ C mÐa idiotim tou L, kai v ∈ V èna idiodi�nusma gia thn idiotim λ,

L(v) = λv. Tìte

〈v, v〉 = 〈L(v), L(v)〉 = 〈λv, λv〉 = λλ〈v, v〉 .

AfoÔ 〈v, v〉 6= 0, λλ = 1.

�


Di�lexh 20

DiagwniopoÐhsh ermitian¸n telest¸n.

L mma 4.6 (L mma Schur.) JewroÔme dianusmatikì q¸ro V peperasmènhc di�sta-

shc, me eswterikì ginìmeno p�nw ap to C, kai grammikì telest L : V → V . Tìte up�rqei

orjokanonik b�sh tou V wc proc thn opoÐa o L èqei �nw trigwnikì pÐnaka.

Apìdeixh. H apìdeixh akoloujeÐ ta b mata tou Jewr matoc TrigwnopoÐhshc, Je¸rh-

ma 1.23. Prèpei na deÐxoume oti e�n V eÐnai q¸roc me eswterikì ginìmeno, mporoÔme na

epilèxoume th b�sh B′ na eÐnai orjokanonik .

Upojètoume oti dimV = n ≥ 2. O telest c L èqei mÐa idiotim λ1. 'Estw u1 èna idiodi�-

nusma me ||u1|| = 1. Sumplhr¸noume se orjokanonik b�sh tou V , B = {u1, u2, . . . , un}kai jewroÔme ton pÐnaka A = BLB. H pr¸th st lh tou A eÐnai to di�nusma suntetagmènwn

tou L(u1) = λ1u1, kai sunep¸c o A èqei th morf

A =

λ1 a12 . . . a1n

0... D

0

.

JewroÔme to V wc eujÔ �jroisma twn V1 = 〈u1〉 kai U = 〈u2, . . . , un〉 kai tic kanonikècapeikonÐseic j2 : U → V1 ⊕ U kai p2 : V1 ⊕ U → U , sel. 62. OrÐzoume thn apeikìnish

M = p2 ◦ L ◦ j2 : U → U

kai parathroÔme oti o pÐnakac pou parist�nei thn apeikìnishM wc proc th b�sh {u2, . . . , un}eÐnai o D.

Apì thn epagwgik upìjesh, up�rqei b�sh W = {w2, . . . , wn}, wc proc thn opoÐa o

pÐnakac T thc apeikìnishc M eÐnai �nw trigwnikìc. ParathroÔme oti o pÐnakac tou L wc

proc th b�sh B′ = {u1, w2, . . . , wn} èqei th morf

B =

λ1 b1 2 . . . b1n

0... T

0

,

dhlad eÐnai �nw trigwnikìc.

�

Pìrisma 4.7 E�n V eÐnai dianusmatikìc q¸roc p�nw apì to R kai to qarakthristikì

polu¸numo χL tou telest L : V → V analÔetai se par�gontec pr¸tou bajmoÔ p�nw apì

to R, tìte up�rqei orjokanonik b�sh wc proc thn opoÐa o L èqei �nw trigwnikì pÐnaka.


Apìdeixh. AfoÔ χL(x) = (−1)n(x−λ1) · · · (x−λn), oi idiotimèc tou L eÐnai oi pragmatikoÐ

arijmoÐ λ1, . . . , λn, kai up�rqei toul�qiston èna idiodi�nusma u1 ∈ V , me ||u1|| = 1, èstw

gia thn idiotim λ1.

Gia na efarmìsoume thn epagwg ìpwc sto Je¸rhma, arkeÐ na parathr soume oti χL(x) =

−(x− λ1)χM (x), kai sunep¸c χM epÐshc analÔetai se paragìntec pr¸tou bajmoÔ.

�

Je¸rhma 4.8 (Fasmatikì Je¸rhma) K�je ermitianìc telest c se èna dianusma-

tikì q¸ro peperasmènhc di�stashc me eswterikì ginìmeno èqei mÐa b�sh apì orjog¸nia

idiodianÔsmata. O pÐnakac tou telest wc proc aut th b�sh eÐnai diag¸nioc, me tic (prag-

matikèc) idiotimèc sth diag¸nio.

Apìdeixh. AfoÔ o telest c L eÐnai ermitianìc, oi idiotimèc tou eÐnai pragmatikoÐ arijmoÐ,

kai sunep¸c to qarakthristikì polu¸numo eÐnai ginìmeno paragìntwn pr¸tou bajmoÔ kai

sthn perÐptwsh pou to s¸ma eÐnai oi pragmatikoÐ arijmoÐ.

Apì to L mma tou Schur, up�rqei orjokanonik b�sh wc proc thn opoÐa o pÐnakac A

tou L eÐnai �nw trigwnikìc. Tìte o pÐnakac A∗ eÐnai k�tw trigwnikìc. All� afoÔ o L eÐnai

ermitianìc, A∗ = A, kai sunep¸c o pÐnakac A eÐnai diag¸nioc. Tìte ta diag¸nia stoiqeÐa

eÐnai oi idiotimèc tou telest , en¸ h b�sh apoteleÐtai apì idiodianÔsmata tou telest .

�

Prìtash 4.9 Gia k�je ermitianì pÐnaka A ∈M(n, C) up�rqei orjomonadiaÐoc pÐnakac U

tètoioc ¸ste Λ = U−1AU eÐnai pragmatikìc diag¸nioc pÐnakac.

Gia k�je summetrikì pÐnaka A ∈ M(n, R) up�rqei orjog¸nioc pÐnakac Q tètoioc ¸ste

Λ = Q−1AQ eÐnai pragmatikìc diag¸nioc pÐnakac.

Je¸rhma 4.10 (Je¸rhma Fasmatik c An�lushc.) K�je ermitianìc pÐnakac

A ∈M(n, C) me k diaforetikèc idiotimèc ekfr�zetai wc �jroisma

A = λ1P1 + · · ·+ λkPk ,

ìpou λi, gia i = 1, . . . , k, eÐnai oi idiotimèc kai Pi eÐnai o pÐnakac orjog¸niac probol c ston

idiìqwro thc idiotim c λi.

Apìdeixh. UpenjumÐzoume oti ènac trìpoc na perigr�youme to ginìmeno dÔo pin�kwn,

AB eÐnai wc �jroisma pin�kwn pou prokÔptoun apì to ginìmeno thc i-st lhc tou A me thn

i-gramm tou B. Sugkekrimèna,

a11 . . . a1n...

. . ....

am1 . . . amn

b11 . . . b1k...

. . ....

bn1 . . . bnk

=

n∑i=1

a1i...

ami

[ bi1 . . . bik

].


AnalÔoume me autì ton trìpo to ginìmeno A = U(ΛU∗).u11 . . . u1n...

. . ....

un1 . . . unn

λ1u11 . . . λ1un1

.... . .

...

λnu1n . . . λnunn

=

λ1

u11...

un1

[ u11 . . . un1

]+ · · ·+ λn

u1n...

unn

[ u1n . . . unn

].

ParathroÔme oti u1i...

uni

[ u1i . . . uni

]eÐnai o pÐnakac orjog¸niac probol c ston upìqwro pou par�getai apì to idiodi�nusma

(u1i, . . . , uni).

E�n h idiotim λj èqei pollaplìthta k kai orjokanonik� idiodianÔsmata w1, . . . , wk, tìte

o pÐnakac orjog¸niac probol c ston idiìqwro thc λj eÐnai to �jroisma twn orjogwnÐwn

probol¸n se k�je èna apì ta w1, . . . , wk,

Pj = w1w∗1 + · · ·+ wkw

∗k .

�


Di�lexh 21

KanonikoÐ telestèc

Oi ermitianoÐ den eÐnai oi mìnoi telestèc pou diagwniopoioÔntai apì orjokanonik� idiodianÔ-

smata. Gia telestèc se migadikoÔc dianusmatikoÔc q¸rouc, mporoÔme na diatup¸soume èna

aplì krit rio pou qarakthrÐzei touc orjog¸nia diagwniopoi simouc telestèc.

Orismìc 4.6. JewroÔme dianusmatikì q¸ro V me eswterikì ginìmeno. 'Enac telest c

L : V −→ V lègetai kanonikìc e�n

L ◦ L∗ = L∗ ◦ L .

E�n A eÐnai o pÐnakac tou telest L wc proc mÐa orjokanonik b�sh tou V , tìte o

pÐnakac tou telest L∗ eÐnai o suzug c pÐnakac A∗, kai o telest c L eÐnai kanonikìc e�n

kai mìnon e�n AA∗ = A∗A.

Par�deigma 4.6 JewroÔme ton pÐnaka A =

[2 −3

3 2

]kai ton telest TA : C2 −→ C2.

Tìte A∗ =

[2 3

−3 2

]kai

AA∗ =

[13 0

0 13

]= A∗A ,

�ra TA eÐnai kanonikìc telest c sto C2 me to kanonikì eswterikì ginìmeno. Oi idiotimèc

tou TA eÐnai 2+3i, 2−3i, kai ta idiodianÔsmata 1√2(i, 1), 1√

2(−i, 1) apoteloÔn orjokanonik

b�sh tou C2. Wc proc aut th b�sh, o pÐnakac tou TA eÐnai o

1

2

[−i 1

i 1

][2 −3

3 2

][i −i

1 1

]=

[2 + 3i 0

0 2− 3i

].

L mma 4.11 E�n V eÐnai dianusmatikìc q¸roc me eswterikì ginìmeno, L eÐnai kanonikìc

telest c ston V kai v eÐnai idiodi�nusma gia thn idiotim λ, tìte v eÐnai epÐshc idiodi�nusma

tou telest L∗ gia thn idiotim λ.

Apìdeixh. JewroÔme ton telest L− λI, kai èqoume

(L− λI) ◦ (L− λI)∗ = (L− λI) ◦ (L∗ − λI) = L ◦ L∗ − λL∗ − λL+ |λ|2I ,

(L− λI)∗ ◦ (L− λI) = (L∗ − λI) ◦ (L− λI) = L∗ ◦ L− λL∗ − λL+ |λ|2I ,

kai afoÔ L eÐnai kanonikìc, L− λI eÐnai epÐshc kanonikìc.


Gia to idiodi�nusma v isqÔei (L− λI)(v) = 0. 'Ara

0 = 〈(L− λI)(v), (L− λI)(v)〉 = 〈v, (L− λI)∗ ◦ (L− λI)(v)〉

= 〈v, (L− λI) ◦ (L− λI)∗(v)〉

= 〈(L− λI)∗(v), (L− λI)∗(v)〉 ,

dhlad (L∗ − λI)(v) = 0 kai v eÐnai idiodi�nusma tou L∗ gia thn idiotim λ.

�

L mma 4.12 Ta idiodianÔsmata enìc kanonikoÔ telest gia diaforetikèc idiotimèc eÐnai

orjog¸nia.

Apìdeixh. E�n λ1, λ2 eÐnai dÔo diaforetikèc idiotimèc tou telest L, kai v1, v2 eÐnai

antÐstoiqa idiodianÔsmata,

λ1〈v1, v2〉 = 〈λ1v1, v2〉

= 〈L(v1), v2〉

= 〈v1, L∗(v2)〉

= 〈v1, λ2v2〉 = λ2〈v1, v2〉 .

AfoÔ λ1 6= λ2, sumperaÐnoume oti 〈v1, v2〉 = 0.

�

Je¸rhma 4.13 JewroÔme dianusmatikì q¸ro V peperasmènhc di�stashc dimV = n

p�nw apì to C, me eswterikì ginìmeno, kai telest L : V −→ V . O telest c L eÐnai

kanonikìc e�n kai mìnon e�n o L eÐnai diagwniopoi simoc, dhlad up�rqei orjokanonik

b�sh tou V pou apoteleÐtai apì idiodianÔsmata tou L. Wc proc aut th b�sh, o pÐnakac tou

L eÐnai diag¸nioc.

Apìdeixh. Upojètoume oti up�rqei orjokanonik b�sh tou V apì idiodianÔsmata tou L.

O pÐnakac tou L wc proc aut th b�sh eÐnai

Λ =

λ1 0

. . .

0 λn

,kai o pÐnakac tou suzugoÔc telest L∗ wc proc thn Ðdia b�sh eÐnai

Λ∗ = Λ =

λ1 0

. . .

0 λn

.AfoÔ ΛΛ∗ = Λ∗Λ, èpetai oti L ◦ L∗ = L∗ ◦ L kai o L eÐnai kanonikìc.

AntÐstrofa, upojètoume oti o telest c L eÐnai kanonikìc. O L èqei mÐa idiotim λ1 ∈C, me idiodi�nusma v1. JewroÔme ton upìqwro W1 = 〈v1〉⊥ twn dianusm�twn pou eÐnai


orjog¸nia proc to v1, kai jètoume V1 = W1. E�n w ∈ V1, tìte

〈v1, L(w)〉 = 〈L∗(v1), w〉 = λ1〈v1, w〉 = 0 .

'Ara L(w) ∈ V1, kai V1 eÐnai analloÐwtoc upìqwroc tou L. 'Ara up�rqei èna idiodi�nusma v2

tou L pou an kei ston upìqwro V1. Me ton Ðdio trìpo deÐqnoume oti o upìqwrocW2 = 〈v2〉⊥

eÐnai analloÐwtoc apì ton L. Jètoume V2 = V1∩W2, kai parathroÔme oti V2 eÐnai upìqwroc

tou V1 kai eÐnai analloÐwtoc apì ton L.

Upojètoume oti gia k < n èqoume breÐ idiodianÔsmata v1, v2, . . . , vk tou L tètoia ¸ste

〈vi, vj〉 = 0 gia i 6= j, kai Vk = 〈v1〉⊥ ∩ · · · 〈vk〉⊥ eÐnai upìqwroc tou V analloÐwtoc apì

ton L. Tìte up�rqei idiodi�nusma vk+1 tou L ston Vk, kai o upìqwroc Vk+1 = Vk ∩〈vk+1〉⊥

eÐnai analloÐwtoc apì ton L.

AfoÔ dimV < ∞, katal goume se b�sh {v1, . . . , vn} tou V , pou apoteleÐtai apì idio-

dianÔsmata tou L. Jètoume qi = 1〈vi, vi〉1/2

vi, kai {q1, . . . , qn} eÐnai orjokanonik b�sh tou

V apì idiodianÔsmata tou L.

�


Di�lexh 22

Efarmog : Kanonik exÐswsh mÐac kwnik c tom c

Sthn Analutik GewmetrÐa (EpÐpedo kai Q¸roc, Kef�laio 4), eÐdame oti h genik exÐswsh

mÐac kwnik c tom c wc proc èna orjokanonikì sÔsthma anafor�c (O, ~i, ~j) sto epÐpedo eÐnai

Ax2 + 2Bxy + Cy2 + 2Dx+ 2Ey + F = 0 . (4.1)

Me thn epilog kat�llhlou sust matoc anafor�c mporoÔme na fèroume thn exÐswsh aut se

mÐa apì tic kanonikèc morfèc thc exÐswshc, gia thn èlleiyh, thn parabol thn uperbol .

Gia thn eÔresh tou kat�llhlou sust matoc anafor�c qrhsimopoioÔme th diagwniopoÐhsh

enìc pÐnaka.

Xekin�me me thn parat rhsh oti h exÐswsh 4.1 mporeÐ na grafteÐ sthn akìloujh morf ,

qrhsimopoi¸ntac dianÔsmata kai pÐnakec[x y

] [ A B

B C

][x

y

]+ 2

[D E

] [ x

y

]+ F = 0 . (4.2)

AfoÔ o pÐnakac S =

[A B

B C

]eÐnai summetrikìc, up�rqei orjokanonik b�sh tou R2

apì idiodianÔsmata tou S. SumbolÐzoume Q ènan 2×2 orjog¸nio pÐnaka pou èqei wc st lec

monadiaÐa idiodianÔsmata tou S. GnwrÐzoume oti Q−1 = QT kai QTSQ eÐnai o diag¸nioc

pÐnakac Λ =

[A′ 0

0 C ′

], me tic idiotimèc A′ kai C ′ tou S sth diag¸nio. 'Etsi S = QΛQT

kai h exÐswsh 4.2 gÐnetai[x y

]Q

[A′ 0

0 C ′

]QT

[x

y

]+ 2

[D E

] [ x

y

]+ F = 0 . (4.3)

Jètoume [x′

y′

]= QT

[x

y

]kai

[D′ E′

]=[D E

]Q

kai èqoume [x′ y′

] [ A′ 0

0 C ′

][x′

y′

]+ 2

[D′ E′

] [ x′

y′

]+ F = 0 . (4.4)

Par�deigma 4.7 JewroÔme thn exÐswsh

3x2 − 10xy + 3y2 + 14x− 2y + 3 = 0 ,

ìpou A = 3, B = −5, C = 3, D = 7, E = −1 kai F = 3. Gr�foume thn exÐswsh sth morf

4.2, [x y

] [ 3 −5

−5 3

][x

y

]+ 2

[7 −1

] [ x

y

]+ 3 = 0 .


Oi idiotimèc tou pÐnaka S =

[3 −5

−5 3

]eÐnai λ1 = 8 kai λ2 = −2. Ta antÐstoiqa monadiaÐa

idiodianÔsmata eÐnai ±v1 kai ±v2, ìpou

v1 =1√2

[1

−1

]kai v2 =

1√2

[1

1

].

Epilègoume thn orjokanonik b�sh tou R2, {v1, v2}. O orjog¸nioc pÐnakac

Q =

[1√2

1√2

− 1√2

1√2

]

diagwniopoieÐ ton pÐnaka S,

S = Q

[8 0

0 −2

]QT .

Antikajistoume sthn 4.2 kai èqoume

[x y

]Q

[8 0

0 −2

]QT

[x

y

]+ 2

[7 −1

] [ x

y

]+ 3 = 0 .

Jètoume [x′

y′

]= QT

[x

y

]opìte [

x y]Q

[8 0

0 −2

]QT

[x

y

]=[x′ y′

] [ 8 0

0 −2

][x′

y′

],

kai h exÐswsh 4.2 gÐnetai

[x′ y′

] [ 8 0

0 −2

][x′

y′

]+ 2

[7 −1

]Q

[x′

y′

]+ 3 = 0 ,

dhlad

8x′2 − 2y′2 + 8√

2x′ + 6√

2y′ + 3 = 0 .

'Etsi èqoume petÔqei ton pr¸to stìqo, na mhdenisteÐ o suntelest c tou miktoÔ ìrou xy.

Gia na broÔme to kèntro summetrÐac thc kampÔlhc, sumplhr¸noume ta tetr�gwna

8

(x′2 +

√2x′ +

1

2

)− 4− 2

(y′2 − 3

√2y′ +

9

2

)+ 9 + 3 = 0 ,

kai èqoume

8

(x′ +

1√2

)2

− 2

(y′ − 3√

2

)2

+ 8 = 0 .

E�n t¸ra jèsoume X = x′ + 1√2kai Y = y′ − 3√

2èqoume thn exÐswsh mÐac uperbol c se

kanonik morf :Y 2

4−X2 = 1 .


O orjog¸nioc pÐnakacQ dra wc mÐa peristrof twn axìnwn, sto sugkekrimèno par�deigma

peristrof kat� gwnÐa −π4 . E�n all�xoume th di�taxh twn dianusm�twn sthn epilegmènh

b�sh, epilèxoume èna apì ta −v1, −v2, tìte o orjog¸nioc pÐnakac ja èqei orÐzousa −1:

h dr�sh tou sto epÐpedo ja eÐnai sÔnjesh mÐac peristrof c kai mÐac an�klashc. O pÐnakac

R =

[1√2

1√2

1√2− 1√

2

]dra wc an�klash sthn eujeÐa y = 0 kai peristrof kat� gwnÐa π

4 .

Oi �xonec sto sÔsthma suntetagmènwn (X, Y ) dÐdontai apì tic exis¸seic Y = 0 kai

X = 0, dhlad tic eujeÐec me exis¸seic 1√2x+ 1√

2y − 3√

2= 0 kai 1√

2x− 1√

2y + 1√

2wc proc

to arqikì sÔsthma anafor�c.

'Otan h exÐswsh mÐac mh ekfulismènhc kwnik c tom c eÐnai se kanonik morf , o tÔpoc thc

kampÔlhc kajorÐzetai apì to prìshmo tou ginomènou twn suntelest¸n twn tetragwnik¸n

ìrwn. E�n oi suntelestèc tou x2 kai tou y2 èqoun to Ðdio prìshmo, h exÐswsh parist�nei

èlleiyh, e�n oi suntelestèc tou x2 kai y2 èqoun antÐjeto prìshmo, h exÐswsh parist�nei

uperbol , en¸ e�n ènac apì touc suntelestèc eÐnai mhdèn, h exÐswsh parist�nei parabol . To

ginìmeno twn suntelest¸n twn tetragwnik¸n ìrwn eÐnai Ðso me thn orÐzousa tou diag¸niou

pÐnaka QTSQ. All� gnwrÐzoume oti oi orÐzousec ìmoiwn pin�kwn eÐnai Ðsec. Sunep¸c

mporoÔme na broÔme ton tÔpo thc kampÔlhc apì thn orÐzousa tou S. MporoÔme epÐshc

na deÐxoume oti mÐa �llh orÐzousa suntelest¸n thc exÐswshc 4.1 kajorÐzei e�n h exÐswsh

parist�nei m ekfulismènh kwnik tom .

Je¸rhma 4.14 JewroÔme thn exÐswsh 4.1.

E�n

∣∣∣∣∣∣∣A B D

B C E

D E F

∣∣∣∣∣∣∣ 6= 0, tìte h exÐswsh parist�nei

1. èlleiyh e�n AC −B2 > 0,

2. parabol e�n AC −B2 = 0,

3. uperbol e�n AC2B < 0.

E�n

∣∣∣∣∣∣∣A B D

B C E

D E F

∣∣∣∣∣∣∣ = 0, to aristerì mèroc thc 4.1 gr�fetai wc ginìmeno poluwnÔmwn

bajmoÔ 1, kai h exÐswsh parist�nei dÔo temnìmenec eujeÐec, dÔo par�llhlec eujeÐec,

mÐa eujeÐa, èna shmeÐo to kenì sÔnolo.

Par�deigma 4.8 JewroÔme thn exÐswsh

x2 − 4xy + 4y2 − 6x− 8y + 5 = 0 .

Gr�foume thn exÐswsh sth morf [x y

] [ 1 −2

−2 4

][x

y

]+ 2

[−3 −4

] [ x

y

]+ 5 = 0 .


O pÐnakac

[1 −2

−2 4

]èqei idiotimèc 0 kai 5, kai antÐstoiqa monadiaÐa idiodianÔsmata

(2√5,

1√5

)kai

(1√5,−2√

5

).

JewroÔme ton orjog¸nio pÐnaka Q =

[1√5

2√5

− 2√5

1√5

], o opoÐoc diagwniopoieÐ ton pÐnaka

suntelest¸n, [1 −2

−2 4

]= Q

[5 0

0 0

]QT .

K�noume thn antikat�stash [x′

y′

]= QT

[x

y

]kai èqoume thn exÐswsh

[x′ y′

] [ 5 0

0 0

][x′

y′

]+

2√5

[−3 −4

] [ 1 2

−2 1

][x′

y′

]+ 5 = 0 ,

dhlad

5x′2 + 2√

5x′ − 4√

5y′ + 5 = 0 .

Sumplhr¸nontac ta tetr�gwna èqoume

5

(x′ +

1√5

)2

− 4√

5

(y′ − 1√

5

)= 0 .

H kampÔlh eÐnai mÐa parabol me kanonik exÐswsh

Y =

√5

4X2 .

To shmeÐo anafor�c tou sust matoc suntetagmènwn (X, Y ), sto opoÐo brÐsketai h ko-

ruf thc parabol c, eÐnai to (x′, y′) = (− 1√5, 1√

5), dhlad (x, y) = (1, 3). Oi �xonec tou

nèou sust matoc anafor�c, Y = 0 kai X = 0, eÐnai oi eujeÐec y′ = 1√5kai x′ = − 1√

5, twn

opoÐwn oi exis¸seic wc proc to arqikì sÔsthma anafor�c eÐnai

2x+ y − 1 = 0 , kai x− 2y + 1 = 0 .

Documents

Grammik Algebra' Iusers.math.uoc.gr/~chrisk/LinAlg1-Notes.pdf · 2019. 5. 18. · Oloc' o mhdenoq¸roc tou A iI onom zetai idioq¸roc tou Agia thn idiotim i. TonÐzoume oti to mhdenikì