17- Matrix Basics 1

8/12/2019 17- Matrix Basics 1

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Advanced

Structural

Analysis

DevdasMenonProfessor

IITMadras

([email protected])

NationalProgrammeonTechnologyEnhancedLearning(NPTEL)

www.nptel.ac.in

Lecture17Module3:

BasicMatrixConcepts


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AdvancedStructuralAnalysis

Modules

1. Reviewofbasicstructuralanalysis 1(6lectures)

2. Reviewofbasicstructuralanalysis 2(10lectures)

3. Basicmatrix

concepts

4. Matrixanalysisofstructureswithaxialelements

5. Matrixanalysisofbeamsandgrids

6. Matrixanalysisofplaneandspaceframes

7. Analysisofelasticinstabilityandsecondordereffects


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Module3:BasicMatrixConcepts

Reviewofmatrixalgebra

Introductiontomatrixstructuralanalysis(forceand

displacementtransformations;stiffnessandflexibilitymatrices;basicformulations;equivalent

jointloads).


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ReviewofBasic

ConceptsinStructuralAnalysis

Matrix

Conceptsand

Methods

Structures

withAxialElements

Beamsand Grids

PlaneandSpace

Frames

ElasticInstability

andSecondorder

AnalysisA

dvanced

Stru

ctural

Analys

is


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1INTRODUCTION

2MATRIX

3VECTOR

4ELEMENTARYMATRIXOPERATIONS

5MATRIXMUTLITPLICATION

6TRANSPOSEOFAMATRIX

MatrixConceptsandAlgebra

7RANKOFAMATRIX

8LINEARSIMULTANEOUSEQUATIONS

9MATRIXINVERSION

10EIGENVALUESANDEIGENVECTORS


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MatricesinStructuralAnalysis

LOADS

(input) STRUCTURE

(system)

RESPONSE?

(output)

A AA AR A

R RA RR R

F k k D

F k k D

LoadVectorFASupportDisplacementsDR

InitialDeformationsD*in

DisplacementVectorDASupportReactionsDRInternalForcesF*

MemberDeformationsD*

Analysissoftwarepackages


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Introduction

Thereisanincreasingtendencyamongmodernstructuralengineerstoleanheavilyonsoftwarepackagesforeverything.

Thisinducesafalsesenseofknowledge,securityandpower.

Thecomputerisindeedapowerfultoolandanassetforanystructuralengineer. Itisdangerous,however,tomakethetoolonesmaster,andtomakeitaconvenientsubstitutefor

humanknowledge,experienceandcreativethinking.

Ref.: Preface to Advanced Structural Analysis


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By definition,amatrix isrectangulararrayofelementsarrangedinhorizontalrowsandverticalcolumns.

Theentriesofamatrix,calledelements,arescalarquantities commonlynumbers,butmayalsobefunctions,operatorsorevenmatrices(calledsub

matrices)themselves.

A A

A

mn a

ij

aij

mn

a11

a12

a13

L a1 n

a21

a22

a23

L a2n

a31

a32

a33

L a3n

M M M M

am1

am2

am3

L amm

order

m n square matrix

a

ij 0 i,j null matrix, O

identity matrix,I

aij 0: i j

aij 1 : i j

diagonal elements

1 0 0

0 1 00 0 1

Matrix?


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square matrix

symmetric matrix (m= n= 6)

aij ajibanded matrix

sparse matrix

a 0 0d b 0e f c

a d e0 b f

0 0 c

lower triangular matrix, L upper triangular matrix, U

sub-matrices

Type of matrix?4 -1

-1 -2

-2

9

9

5

74-1

-2

-2

-1

1

1

00

0

0

000

00

0 0

0 00

0 00

00

0

A a

ij

mn

b

ij

ip

cij

l np

dij

ml pe

ij

ml np

mn

B CD E

Partitioning:


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Avectorisasimplearrayofscalarquantities,typicallyarrangedinaverticalcolumn.

Hence,thevectorcanbevisualisedasamatrixoforderm

1,wherethenumberm iscalledthedimensionofthevector.Thescalarentriesofavectorarecalledcomponents ofthevector.

V V m

vi V m1 vij m1

v1

v2

v3

M

vm

Whatisavector?

Isitatype

of

matrix?

a1 a2 a3 L an row vector


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Wecanvisualizeamultidimensionallinearvectorspace,m,whosedimensionm isgivenbytheminimumnumberoflinearlyindependentvectors(withrealcomponents)requiredtospanthespace.Vectorsaresaidtospanavectorspace,ifthespaceconsistsofallpossiblelinearcombinations ofthosevectors.Anysetofvectorsthatarelinearly

independentandalsospanthevectorspaceiscalledabasisofthevectorspace.

0x

y

z

V

V

2

13

canbevisualisedin3 vectorspaceas V 2i j 3k

unitvectors

10

0

'01

0

and00

1

provideanorthogonalbasisin3 vectorspace.

havingamagnitudeorlength,

V v

i

2

i1

m

14

Asetofvectors,{V1,V2,,Vn},havingthesamedimensionm,issaidtobelinearlyindependentifnolinearcombinationofthem(otherthanthezero

combination)resultsinazerovector;i.e.,c1 = c2 = = cn = 0,if .

ciVii1

n

0


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ELEMENTARYMATRIXOPERATIONS

ScalarMultiplication: A aij mn aij mn A

MatrixAddition: A+B aij mn bij mn aij bij mn

A B a

ij

mn

1 bij mn aij bij mn

A+B B+A

A+ B+C A+B +C

MatrixMultiplication: aij mn bij np cij mpAB =C;i.e.,

L L L L L

L L L L L

ai1 ai2 ai3 L ain

L L L L L

L L L L L

mn

M M b1j

M M

M M b2j

M M

M M M M M

M M bnj

M M

np

M

M

L L cij L

M

M

mp

ith row

jth column jth column

ith row


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A B+C

AB+AC

A BC AB C

B+C A BA+CAAO= O

AI= A

2 13 4

1 2

32

1 42 0

22

0 811 125 4

32

A mn B np C mp

23

1

1 14

2

2

EverycolumnvectorofC isalinear

combinationofthecolumnvectorsofthepremultiplyingmatrixA!

3 1 4

4

2 0

EveryrowvectorofC isalinearcombinationoftherowvectorsofthepostmultiplyingmatrixB!

Matrix multiplication operation does notpossess the property of commutativity;i.e, in general, AB BA


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TRANSPOSEOFAMATRIX

Transposition is an operation in which the rectangular array of the matrix is rearranged(transposed) such that the order of the matrix changes from m nto n m, with therows changed into columns, preserving the order. If the original matrix is A =[aij]mn,then thetranspose ofA, which is denoted as AT, is given by AT = [aij]

T = [aji]nm.

Transposition is an operation in which the rectangular array of the matrix is rearranged(transposed) such that the order of the matrix changes from m nto n m, with therows changed into columns, preserving the order. If the original matrix is A =[aij]mn,then thetranspose ofA, which is denoted as AT, is given by AT = [aij]

T = [aji]nm.

(AT)T =A

(A)T =AT

(A+B)T =AT+BT

(AB)T =BTAT

ST =AT (AT)T

A nm

T

A mn S nn

Square matrix

=AT A=S

1 2 3

2 0 1

1 2

2 03 1

14 1

1 5

Symmetric matrix

(sji= sij)

AT=AT (i.e., aji= aji ) Skew - Symmetric


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Theproduct{F}T{D}= resultsinamatrixoforder1 1 ,whichisnothing

butascalar. Suchaproduct,whichissometimesdenotedas ,iscalledinnerproduct (dotproductinvectoralgebrainvolvingtwoandthreedimensionalvectors). Thisproductessentiallyreflectstheprojectedlengthofonevectoralongthedirectionoftheothervector. ThemagnitudeofanyvectorV,defined as maybeviewedasthe

innerproductofVwithitself,i.e., .

Theproduct{F}T{D}= resultsinamatrixoforder1 1 ,whichisnothing

butascalar. Suchaproduct,whichissometimesdenotedas ,iscalledinnerproduct (dotproductinvectoralgebrainvolvingtwoandthreedimensionalvectors). Thisproductessentiallyreflectstheprojectedlengthofonevectoralongthedirectionoftheothervector. ThemagnitudeofanyvectorV,defined as maybeviewedasthe

innerproductofVwithitself,i.e., .

F,D

V vi

2

i1m

V,V VTVorthonormal

vectors

X

i

T

X

j ij 1 if i= j

0 if i j

D,F DT k D

D,F T DT k D T DT k T D [k] is symmetric !

Commutative propertyof inner product: F

T

D D T

F

F

m1 k mm D m1


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Therankofthematrix[A]isequaltothenumberoflinearly

independentcolumnvectorsofthematrix,andthisnumberisidenticaltothenumberoflinearlyindependentrowvectors.

Themaximumvalueoftherankrofanymatrixoforderm n isgivenbym orn(whicheverislower),andtheminimumvalueis1.

Linearsimultaneousequations:

aij

Xj

j1n ci i 1,2,.....,m

A mn Xn1 Cm1coefficientmatrix

RANKOF

AMATRIX

Thesubspaceinthevectorspacem containingalllinearcombinationsoftheindependentcolumn (orrow)vectorsiscalledthecolumnspace

(orrowspace)ofA,andthissubspacehasadimensi0nequaltorankr.

A 1 2 3

2 1 3

3 1 4

4 2 6

Rank of A = 3 or 2 or 1 ?

Sum of first two columns! Rank = 2.

C=O(homogeneousequations)

nullspaceofA(allpossible

solutionsofX)


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RowReducedEchelonForm

R mn

Irr F r nr

O mr rO mr nr

ArelativelyeasyandcertainwayofdeterminingtherankofamatrixisbyreducingthematrixtoarowreducedechelonformR throughaprocessofelimination(transformingA ascloselyaspossibletoanidentifymatrixI intheupperleftcorner).

Free variablecoefficient matrix

A 2 4 6

2 1 3

3 1 4

4 2 6

1 2 3

0 1 1

0 0 0

0 0 0

pivot

1 2 3

0 3 3

0 5 5

0 6 6

1 0 1

0 1 1

0 0 0

0 0 0

R I FO O

Rank r= 2

2 4 6

2 1 3

3 1 4

4 2 6

X1

X2

X3

c1

c2

c3

c4

AX=C:


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LINEARSIMULTANEOUSEQUATIONS

AX=C

I rr F r nrO mrr O mr nr

mn

Xpivot

r1X

free nr1

n1

Dpivot

r1D

zero mr1

m1

pivotvariables pivotrowconstants

freevariables zerorowconstants

Xpivot Dpivot F Xfree X Xp XnD

zero O

Case1 : r m and r n

must be satisfied by C for a feasible solution set.

2 4 6

2 1 3

3 1 4

4 2 6

X1

X2

X3

c1

c2

c3

c4

AX=C

RX=D

1 0 1

0 1 1

0 0 0

0 0 0

X1

X2

X3

c14c2 6c1 c

2 3c

3 c

110c

2 62c2

c4

RX=D

zero

zero

c3

10c2c

1

6

c4 2c2

particular + null space solution


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2 4 6

2 1 3

3 1 4

4 2 6

X1

X2

X3

a

b

(10ba) / 62b

AX=C

2 4 6

2 1 3

3 1 4

4 2 6

X1

X2

X3

0

3

5

6

1 2 3

0 3 3

0 5 5

0 6 6

X1

X2

X3

0356

1 0 1

0 1 1

0 0 0

0 0 0

X1

X2

X3

210

0

For a feasible solution space

X

pivot Dpivot F XfreeD

zero OParticularsolution:LetX3=3

X1 2 X

3

X2 1 X

3

X1

X2

X3

p

210

Nullspacesolution(C=O);LetX3=

X1

X2

X3

n

Complete

solution:

X=Xp+

Xn

X1

X2

X3

n

21


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CoefficientmatrixAhasafullcolumnrank(r=n),buttherearelinearlydependentrows (rn).Wehavemoreequationsthanunknowns,andweneedtoensurethattheequationsareconsistent(linearlydependent)forasolutiontobepossible;i.e., hastobesatisfied.

Case2: r n m

D

zero 0

XD

pivot

R IO

R mn Irr F

r nr

O mr rO mr nr

IrrO mr r

X

pivot Dpivot F XfreeD

zero O must be satisfied by C for a feasible solution set.

A | C 2 4 6 c12 1 3 c

2

3 1 4 c3

4 2 8 c4

1 2 3 c1 / 20 3 3 c

1 c

2

0 5 5 3c1 / 2 c

3

0 6 4 2c1 c

4

shouldbezeroshouldbezero

1 0 1 c

1 4c

2 60 1 1 c1 c2 30 0 2 2c

2 c

4 0 0 0 c

3 c

1 10c

2 6

(rows 3 and 4 interchanged)


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2 4 6

2 1 3

3 1 4

4 2 8

X1

X2

X3

a

b

(10ba) / 6c

AX=C

1 0 0

0 1 0

0 0 1

0 0 0

X1

X2X

3

210

0

X D

X1

X2

X3

p

210

For a feasible solution spacec

3 c

110c

2 6

RXDI

O

X

O

D

O

2 4 6

2 1 3

3 1 4

4 2 8

X1

X2X

3

0

3

5

6


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Case3: r m n

R I F

R mn Irr F r nr

O mr rO mr nr

Irr F r nr

CoefficientmatrixAhasafullrowrank(r=m),buttherearelinearlydependentcolumns(rm).Wehavemoreunknownsthanequations,whichisasituationweencounterinstaticallyindeterminatestructures.

Owingtotheabsenceofzerorowvectors,thereisnoconstraintontheconstantvectorC,andasolutioniscertainlypossible.

However,astherearefreevariablespresent,thenullspacesolutionhasinfinitepossibilities(asinCase1)inthecompletesolution.

X

pivot Dpivot F Xfree X Xp XnD

zero O


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1 2 3 4

2 1 1 2

3 3 4 8

X1

X2

X3

X4

c1

c2

c3

1 2 3 40 3 5 60 3 5 4

X1

X2

X3

X4

c12c

1 c

2

3c1 c

3

1 0 1 3 00 1 5 3 20 0 0 2

X1

X2

X3

X4

c12c

2 32c

1 c

2 3c1 c2 c3

AX=C

1 0 0 1 30 1 0 5 30 0 1 0

X1

X2

X4

X3

c12c2 35c12c

2 3 c3c1 c

2 c

3 2

d1d

2

d3

Interchange third and fourth columns to preserve the identity matrix on the left. I F %X D

RX=D

c1

c2

c3

3

62

d1

d2

d3

37

3.5


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It is evident that infinite solutions are possible.

1 2 3 42 1 1 2

3 3 4 8

X1

X2

X3

X4

c1

c2

c3

AX=C

Xpivot Dpivot F XfreeParticularsolution:LetX3=0

X1

X2

X3

X4

p

3703.5

Nullspacesolution(D=O);LetX3=

X1X

2

X3

1 / 35 / 30

I F %X D

1 0 0 1 30 1 0 5 3

0 0 1 0

X1

X2

X4

X3

377 2

X XpX

n 37

03.5

35 3

0

3 375 3

3.5

Completesolution:


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Inthiscase,thecoefficientmatrixAhasafullcolumnrank(r=n)aswellasafullrowrank (r=m),whichalsoimpliesthatthematrixisasquarematrix(m=n).Suchamatrixissaidtobeinvertibleor nonsingular.

Case4 : r m n

R I unique solution, XDEliminationTechniqueforsolvingAX=C

InthetraditionalGausseliminationprocedure,itissufficienttoreducethecoefficientmatrixAtoanuppertriangularformUforthispurpose,whilecarryingouttheelementarymatrixoperationsontheaugmentedmatrixIfAis asquarematrixandtheequationsareconsistent,theuniquesolutioncanbeobtainedbybacksubstitution.

However,bygoingafewstepsfurther,theAmatrixcanbereducedtotherowreducedechelonformR,andthecompletesolution,ifany,canbedirectlyobtained.

A | C

R I FO O


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WhenthematrixAissquareandoffullrank,analternativeapproachtosolvingtheequationsisbytheoperationofinversion.WhenthematrixAissquareandoffullrank,analternativeapproachtosolvingtheequationsisbytheoperationofinversion.

AX = CMATRIXINVERSION

X A 1 C

AA1 A1A I

BasicPropertiesofInverseofaMatrix:

A1 1 AA T 1 A 1 TA1 1 A 1AB1 B1A1

DeterminantofaMatrix

Ingeneral,itcanbeprovedthattheinverseexists,ifascalarproperlycalled

thedeterminantofthesquarematrixA,denotedas ordetA or ,isnotequaltozero.

Foradiagonalmatrix,thedeterminantisgivenbytheproductofallthe

diagonalelements.

A A


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Perhapsthesimplestwayoffindingthedeterminantofanysquarematrixisby

applyingtheprocessofelimination,reducingthematrixtoanupperdiagonalform U.Thedeterminantisdirectlyobtainableastheproductofthepivotelementsinthediagonal (ve signifoddno.ofrowexchangesinvolved).

For a matrix A of order 2 2, the determinant is given by:

det A a11 a12a

21 a

22

a11

a22 a

21a

12

Clearly,anonzerodeterminantispossibleforagivensquarematrixAonlyiftherearepivotelementsinalltherowsofthematrix;i.e.,thematrixhastohaveafullrankforittobenonsingularandinvertible.

det A

aij

ijj1

n

for i 1,2,3,...,nwhere isthecofactoroftheelement aij,whosevalueisgivenbythedeterminant ijofthe(n1) (n1)matrixobtainedbydeletingthei

th rowandjth

columnofthematrixA,withtheappropriatesign(positiveornegative).

ij 1ij ij


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det A a

11

11 a

12

12 ...a

1 n

1 n

det A a11

a22

a23

a32

a33

a12

a21

a23

a31

a33

a13

a21

a22

a31

a32

det A m det AmdetA n det A for A nndet A

T detAAdjointMethodofFindingInverse

Ingeneral,foranysquarematrixofordern n,provided,det A 0,itcanbeshownthat

where iscalledtheadjointmatrixofA,whichisthetransposeofamatrixwhoseelementscomprisethecofactorsijofA.Thistechnique,however,becomescumbersomewhentheorderofthematrixexceedsthreeorfour.

A1 1detA

A


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a b

c d

1 1

adbc d bc a

a sym

b c

d e f

1

1a cf e2 b bf ded be cd

cf e2 symdebf fad2be cd bdae acb2

7 3 2 3 3 82 3 7 3 3 83 8 3 8 3 8

1

11.40625

0.734375 0.109375 0.6250.109375 0.734375 0.6250.625 0.625 5.0

Algorithmbasediterativemethodsoffindinginverse:

GaussJordanEliminationMethod LDLTDecompositionMethod Cholesky DecompositionMethod


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Cramersruleissuitableforsolvingasmallnumberofsimultaneousequations.

Itrequiresgenerationofn+1determinants,whichiscumbersomebyalgebraicformulation.Eliminationbasedalgorithmicmethodsaremuchbettersuitedforcomputerapplication.

CramersRule

Thesolutiontoasetofconsistentequations,canbeshowntobegivenby:

A X C

Xi

a11

a1 ,j1 c1 a1 n

Ma

n1 a

n,j1 cn anna

11 a

1 ,j1 a1j a1 nM

an1

an,j1 anj ann

i 1,2,.....,n


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Thestabilityofiterativesolutionprocedures(formatrixinversion)andtheaccuracyoftheendresultdependontheconditionofthematrix,whichisameasureofitsnonsingularity.Amatrixissaidtobeillconditioned(ornearsingular)ifitsdeterminantisverysmallincomparisonwiththevalueofits

averageelement,andsuchamatrixisvulnerabletoerroneousestimationofitsinversebyiterativesolvers.

Stiffnessmatricesarerelativelywellconditionedandhavethepropertyofpositivedefiniteness,withdiagonaldominance,wherebytheirinversescanbestablyandaccuratelygenerated.

ConditionofaMatrix

ThesquarematrixAofordernissaidtobepositivedefinite,ifforanyarbitrarychoiceof

anndimensionalvectorX,theproductXT

AXyieldsascalarquantitythatisinvariablypositive.Alltheeigenvaluesofsuchamatrixarereal,distinctandspacedwellapart.

ThesquarematrixAofordernissaidtobepositivedefinite,ifforanyarbitrarychoiceof

anndimensionalvectorX,theproductXT

AXyieldsascalarquantitythatisinvariablypositive.Alltheeigenvaluesofsuchamatrixarereal,distinctandspacedwellapart.

XTAX 0

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17- Matrix Basics 1