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2664

3!1!6

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3775

2664

1001

3775

2664

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3775

2664

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2664

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3775

2664

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3775

266664

20

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0

377775

266664

402

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377775

266664

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377775

266664

848

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377775

266664

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266664

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377775

266664

30

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377775

266664

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266664

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377775

266664

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377775

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24

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2664

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f 1; 2; 3g W

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f 1; : : : ; pg f 1; : : : ; pgk 1; : : : ; k

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W D f 1; 2; 3g f 1; 2; 3gf 1; 2; 3g W

f 1; 2; 3g W

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A D QR Q m ! n R n ! nA R

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k # A kk # A k

k # A k

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k # AOk % k # A kRn

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A D

24

5 !4 !2!4 5 2!2 2 2

35

1 D

24

!221

35

2 D

24

110

35

1 2 AA

n " n

AT D A A D 3A D 4 ! D 0

n " n n

Rn T

B D T P T D P !1 DB

A n " n B n " mBTAB BTB BBT

A n " n A /! D!.A / ; Rn

AA!1

A BAB D BA AB

A D !1 P D! A k

! k D! k

A D PRP !1 P RA R

A

A

Rn B D T

Rn B B

B B2 D B

B

B n " n B2 B

Rn O D BD ! O

OW B

W W ?

BB

!.A ! !I /

2664

5 2 9 !62 5 !6 99 !6 5 2

!6 9 2 5

3775

2664

:38 !:18 !:06 !:04!:18 :59 !:04 :12!:06 !:04 :47 !:12!:04 :12 !:12 :41

3775

2664

:31 :58 :08 :44:58 !:56 :44 !:58:08 :44 :19 !:08:44 !:58 !:08 :31

3775

266664

10 2 2 !6 92 10 2 !6 92 2 10 !6 9

!6 !6 !6 26 99 9 9 9 !19

377775

.A2/T D .AA/T D ATAT AT D A

.A2/T D AA D A2 A2

A AA2

D P

Q. / D TA D T D D !1y21 C !2y2

2 C # # # C !ny2n

!1; : : : ;!n A PQ. /

¤!1; : : : ;!n

Q. / D 3x21 C 2x2

2 C x23 C 4x1x2 C 4x2x3

Positive definite

z

Indefinite

z

Negative definite

z

x1 x2

x1

x1x2

x2

A D

24

3 2 02 2 20 2 1

35

A !1 Q

AWEBTA

AA A D RTR R

A

A

WEB

TA A D!

5 1=31=3 1

"

D!

x1

x2

"D

!61

"D

!13

"

TA A D

24

4 3 03 2 10 1 1

35

D

24

x1

x2

x3

35 D

24

2!1

5

35 D

264

1=p

3

1=p

3

1=p

3

375

R2

10x21 ! 6x1x2 ! 3x2

2 5x21 C 3x1x2

R2

20x21 C 15x1x2 ! 10x2

2 x1x2

R3

8x21 C 7x2

2 ! 3x23 ! 6x1x2 C 4x1x3 ! 2x2x3

4x1x2 C 6x1x3 ! 8x2x3

R3

5x21 ! x2

2 C 7x23 C 5x1x2 ! 3x1x3

x23 ! 4x1x2 C 4x2x3

D Px2

1 C 10x1x2 C x22

P

A

9x21 C 7x2

2 C 11x23 ! 8x1x2 C 8x1x3

AP

D P TAP

D P

P

3x21 ! 4x1x2 C 6x2

2 9x21 ! 8x1x2 C 3x2

2

2x21 C 10x1x2 C 2x2

2 !5x21 C 4x1x2 ! 2x2

2

x21 ! 6x1x2 C 9x2

2 8x21 C 6x1x2

!2x21 ! 6x2

2 ! 9x23 ! 9x2

4 C 4x1x2 C 4x1x3 C 4x1x4 C6x3x4

4x21 C 4x2

2 C 4x23 C 4x2

4 C 3x1x2 C 3x3x4 ! 4x1x4 C4x2x3

x21 C x2

2 C x23 C x2

4 C 9x1x2 ! 12x1x4 C 12x2x3 C 9x3x4

11x21 ! x2

2 ! 12x1x2 ! 12x1x3 ! 12x1x4 ! 2x3x4

5x21 C 8x2

2 D .x1; x2/T D 1

x21 C x2

2 D 1

5x21 ! 3x2

2T D 1

n " n Rn

TAA

Q Q. / > 0Rn

ATA

AA D RTR R

k k2

A PD P TA

A 2 " 2TA D c c

A TA¤ A

Q. / D TA A D!

a bb d

"A ¤ 0

A

!1 !2 AA .A ! !I /

.! ! !1/.! ! !2/ !1 C !2 Da C d A !1!2 D A

Q A > 0 a > 0

Q A > 0 a < 0

Q A < 0

B m " n BTBB n " n BTB

n " n AB A D BTB

A D T P T D P !1

C D D C TC B D PCP T

B

A B n " nA C B

A n " nTA

TA!1

SG

D P

TA D T D D !1y21 C !2y

22 C # # # C !ny2

n

!iTA

D i i In

TAPositive semidefinite

z

x1 x2

Q. /

Rn

k k D 1; k k2 D 1; T D 1

x21 C x2

2 C # # # C x2n D 1

T D 1Q

Q. / T D 1

Q. / D 9x21 C 4x2

2 C 3x23

T D 1

x22 x2

3

4x22 % 9x2

2 3x23 % 9x2

3

Q. / D 9x21 C 4x2

2 C 3x23

% 9x21 C 9x2

2 C 9x23

D 9.x21 C x2

2 C x23/

D 9

x21 C x2

2 C x23 D 1 Q. /

Q. / D 9 D .1; 0; 0/Q. / T D 1

Q. /

9x21 $ 3x2

1; 4x22 $ 3x2

2

Q. / $ 3x21 C 3x2

2 C 3x23 D 3.x2

1 C x22 C x2

3/ D 3

x21 C x2

2 C x23 D 1 Q. / D 3 x1 D 0 x2 D 0 x3 D 1

Q. / T D 1

y

Parks andrecreation

1.4

4x2 + 9y2 = 36(indifference curves)

q (x, y) = 4q (x, y) = 3

q (x, y) = 2Road and bridge repair

2.1x

.2:1; 1:4/

x1 D x

3; x2 D y

2; ; x D 3x1 y D 2x2

x21 C x2

2 D 1

q.3x1; 2x2/ D .3x1/.2x2/ D 6x1x2 D!

x1

x2

"

Q. / D 6x1x2T D 1 Q. / D

TA

A D!

0 33 0

"

A ˙3

"1=

p2

1=p

2

#! D 3

"!1=

p2

1=p

2

#

! D !3 Q. / D q.x1; x2/ x1 D 1=p

2x2 D 1=

p2

x D 3x1 D3=

p2 & 2:1 y D 2x2 D

p2 & 1:4

q.x; y/ D 3.x; y/

Q. / D 3x21 C 3x2

2 C 2x1x2 Q

Q Q. /T D 1

D PTA TD

5x21 C 6x2

2 C 7x23 C 4x1x2 ! 4x2x3 D 9y2

1 C 6y22 C 3y2

3

3x21 C 2x2

2 C 2x23 C 2x1x2 C 2x1x3 C 4x2x3 D 5y2

1 C 2y22

y23

Q. /T D 1

Q. /T D 1 T D 0

Q. / D 5x21 C 6x2

2 C 7x23 C 4x1x2 ! 4x2x3

Q. / D 3x21 C 2x2

2 C 2x23 C 2x1x2 C 2x1x3 C 4x2x3

Q. / D 5x21 C 5x2

2 ! 4x1x2

Q. / D 7x21 C 3x2

2 C 3x1x2

Q. / D !2x21 ! x2

2 C 4x1x2 C 4x2x3

R3 Q. / T D 1

Q !1 !4

Q. / D 7x21 C x2

2 C 7x23 ! 8x1x2 ! 4x1x3 ! 8x2x3

R3 Q. /T D 1

Q !3

Q. / D 7x21 C 3x2

2 ! 2x1x2

x21 C x2

2 D 1

Q. / D !3x21 C 5x2

2 ! 2x1x2

x21 C x2

2 D 1

ATA

! Am % ! % M

m M! D TA

A n " n M m

TA 1

n tM m t D TA

t D .1 ! ˛/m C ˛M ˛D

p1 ! ˛ n C p

˛ 1T D 1 TA D t

x1x2 C 3x1x3 C 30x1x4 C 30x2x3 C 3x2x4 C x3x4

3x1x2 C 5x1x3 C 7x1x4 C 7x2x3 C 5x2x4 C 3x3x4

4x21 ! 6x1x2 ! 10x1x3 ! 10x1x4 ! 6x2x3 ! 6x2x4 ! 2x3x4

!6x21 ! 10x2

2 ! 13x23 ! 13x2

4 ! 4x1x2 ! 4x1x3 ! 4x1x4 C6x3x4

A D!

3 11 3

"

x

4

z

x1

x2

!""""""#""""""$

Q. /T D 1

"1=

p2

1=p

2

# "!1=

p2

1=p

2

#

D P P D"

1=p

2 !1=p

2

1=p

2 1=p

2

#

TD D 4y21 C 2y2

2

Q. /!1=

p2

1=p

2

" !10

"

TD Q. /

A D PDP !1

D A D QDP !1 m " nA

A A

A D U †V T ATWEB

A AT

!1 00 !3

" !!5 0

0 0

"

! p6 1

0p

6

" ! p3 2

0p

3

"

U

24

!1=3 2=3 2=32=3 !1=3 2=32=3 2=3 !1=3

35

U

264

1=p

6

!2=p

6

1=p

6

375

!!3 0

0 0

" !!2 0

0 !1

"

!2 !12 2

" !2 30 2

"

24

7 10 05 5

35

24

4 !22 !10 0

35

24

!3 16 !26 !2

35

24

1 10 1

!1 1

35

A D!

3 2 22 3 !2

"AT

A

AU V

A D

24

:40 !:78 :47:37 !:33 !:87

!:84 !:52 !:16

35

24

7:10 0 00 3:10 00 0 0

35

"

24

:30 !:51 !:81:76 :64 !:12:58 !:58 :58

35

A

AA AV

3 " 4A

A D

24

!:86 !:11 !:50:31 :68 !:67:41 !:73 !:55

35

24

12:48 0 0 00 6:34 0 00 0 0 0

35

"

2664

:66 !:03 !:35 :66!:13 !:90 !:39 !:13

:65 :08 !:16 !:73!:34 :42 !:84 !:08

3775

A m " nA D U †V T U m " m

† m " n rV n " n

AA!1

A j AjA

V ATAU AAT

† AATA AAT

A n " nA D T

A

P m " m PAA

A kA k1 1

A

U D Π1 # # # m " V D Π1 # # # n "

i i

A D #1 1T1 C #2 2

T2 C # # # C #r r

Tr :

ATj D #j j

1 % j % r D A

T W Rn ! Rm

B Rn C Rm

T B C m " n

A D

2664

!18 13 !4 42 19 !4 12

!14 11 !12 8!2 21 4 8

3775

A D

2664

6 !8 !4 5 !42 7 !5 !6 40 !1 !8 2 2

!1 !2 4 4 !8

3775

4 " 4

#1=#4

5 " 5

#1=#5

A D U †V T † m " n AT D .V T /T †T U T D V †T U T

AT V U †T n " m† †T A AT

A 2 " n AAT 2 " 2ATA

Rn

R8 8 " 300

N j R2

j w h

A n " n

A A

A A

A kA k D k kRn

TAP A

P n " nP T D P !1

TA > 0 TA

TA k k D 1A

TA A

D P P

P n " nD P TA

P !1

U m " n U U T

U

B m " n Rn kB k % #1

#1 B

m " n BB D P †Q P m " mQ n " n

† m " n

A n " n A ATA

f 1; : : : ; ng Rn

!1; : : : ;!n

A D !1 1T1 C # # # C !n n

Tn

A

!1; : : : ;!n A

A n " n rA A

r 1

A n " n

. A/? D A

Rn DO C O A A

n " n AA A

A n " nRn

n " n AA A D RTR

R

AA A D LU U

A m " n G D ATAA G

A

AA

n " n Gr G r " n

A GG G

BBT n " r B

n " n AA D PQ P n " n

A Q n " n

A D U †V T A D .U †U T /.UV T /

PP Q

m " n AA D UrDV T

r

AC D VrD!1U T

r

AC

Rm AAC

A

Rn ACAA

AACA D A ACAAC D AC

A DC D AC

A A DC D C

A DC A A

C A DA D

k Ck % k k D C

Rm AC

A D O OA

AA

ATAA D D .6; !1; !4; 6/

O OA A

kOk < kO C k

A D

2664

!3 !3 !6 6 1!1 !1 !1 1 !2

0 0 !1 1 !10 0 !1 1 !1

3775

A D

2664

4 0 !1 !2 0!5 0 3 5 0

2 0 !1 !2 06 0 !3 !6 0

3775