eciv 520 structural analysis ii review of matrix algebra

ECIV 520

Structural Analysis II

Review of Matrix Algebra

Linear Equations in Matrix Form

10z8y3x5

6z3yx12

24z23y6x10

Linear Equations in Matrix Form

10z8y3x5

10

z

y

x

835

Linear Equations in Matrix Form

6

z

y

x

3112

6z3yx12

Linear Equations in Matrix Form

24

z

y

x

23610

24z23y6x10

23610

3112

835

z

y

x

24

6

10

10

z

y

x

835

6

z

y

x

3112

24

z

y

x

23610

Matrix Algebra

nm3n2n1n

m3333231

m2232221

m1131211

aaaa

aaaa

aaaa

aaaa

A

Rectangular Array of Elements Represented by a single symbol [A]

Matrix Algebra

nm3n2n1n

m3333231

m2232221

m1131211

aaaa

aaaa

aaaa

aaaa

A

Row 1

Row 3

Column 2 Column m

n x m Matrix

nm3n2n1n

m3333231

m2232221

m1131211

aaaa

aaaa

aaaa

aaaa

A

Matrix Algebra

32a

3rd Row

2nd Column

Matrix Algebra

m321 bbbbB

1 Row, m Columns

Row Vector

B

Matrix Algebra

n

3

2

1

c

c

c

c

C

n Rows, 1 Column

Column Vector

C

Matrix Algebra

5554535251

4544434241

3534333231

2524232221

1514131211

aaaaa

aaaaa

aaaaa

aaaaa

aaaaa

A

If n = m Square Matrix

e.g. n=m=5e.g. n=m=5Main Diagonal

Matrix Algebra

9264

2732

6381

4215

A

Special Types of Square Matrices

Symmetric: aSymmetric: aijij = a = ajiji

Matrix Algebra

9000

0700

0080

0005

A

Diagonal: aDiagonal: aijij = 0, i = 0, ijj

Special Types of Square Matrices

Matrix Algebra

1000

0100

0010

0001

I

Identity: aIdentity: aiiii=1.0 a=1.0 aijij = 0, i = 0, ijj

Special Types of Square Matrices

nm

m333

m22322

m1131211

a000

aa00

aaa0

aaaa

A

Matrix Algebra

Upper TriangularUpper Triangular

Special Types of Square Matrices

nm3n2n1n

333231

2221

11

aaaa

0aaa

00aa

000a

A

Matrix Algebra

Lower TriangularLower Triangular

Special Types of Square Matrices

nm

3332

232221

1211

a000

0aa0

0aaa

00aa

A

Matrix Algebra

BandedBanded

Special Types of Square Matrices

Matrix Operating Rules - Equality

nm3n2n1n

m3333231

m2232221

m1131211

aaaa

aaaa

aaaa

aaaa

A

pq3p2p1p

q3333231

q2232221

q1131211

bbbb

bbab

bbbb

bbbb

B

[A]mxn=[B]pxq

n=p m=q aij=bij

Matrix Operating Rules - Addition

nm3n2n1n

m3333231

m2232221

m1131211

aaaa

aaaa

aaaa

aaaa

A

pq3p2p1p

q3333231

q2232221

q1131211

bbbb

bbab

bbbb

bbbb

B

[C]mxn= [A]mxn+[B]pxq

n=p

m=qcij = aij+bij

Matrix Operating Rules - Addition

Properties

[A]+[B] = [B]+[A]

[A]+([B]+[C]) = ([A]+[B])+[C]

Multiplication by Scalar

nm3n2n1n

m3333231

m2232221

m1131211

gagagaga

gagagaga

gagagaga

gagagaga

AgD

Matrix Multiplication

nm3n2n1n

m3333231

m2232221

m1131211

aaaa

aaaa

aaaa

aaaa

A

pq3p2p1p

q3333231

q2232221

q1131211

bbbb

bbab

bbbb

bbbb

B

[A] n x m . [B] p x q = [C] n x q

m=p

n

1kkjikij bac

Matrix Multiplication

nm3n2n1n

m3333231

m2232221

m1131211

aaaa

aaaa

aaaa

aaaa

A

pq3p2p1p

q3333231

q2232221

q1131211

bbbb

bbab

bbbb

bbbb

B

1nn13113

2112111111

baba

babac

11c

C

nm3n2n1n

m3333231

m2232221

m1131211

aaaa

aaaa

aaaa

aaaa

A

Matrix Multiplication

pq3p2p1p

q3333231

q2232221

q1131211

bbbb

bbab

bbbb

bbbb

B

3nn23323

2322132123

baba

babac

23c

C

Matrix Multiplication - Properties

Associative: [A]([B][C]) = ([A][B])[C]

If dimensions suitable

Distributive: [A]([B]+[C]) = [A][B]+[A] [C]

Attention: [A][B] [B][A]

nmm3m2m1

3n332313

2n322212

1n312111

T

aaaa

aaaa

aaaa

aaaa

A

nm3n2n1n

m3333231

m2232221

m1131211

aaaa

aaaa

aaaa

aaaa

A

Operations - Transpose

Operations - Trace

5554535251

4544434241

3534333231

2524232221

1514131211

aaaaa

aaaaa

aaaaa

aaaaa

aaaaa

A

Square Matrix

tr[A] = tr[A] = aaiiii

Determinants

nnn2n1

2n2221

1n1211

aaa

aaa

aaa

A

nnn2n1

2n2221

1n1211

aaa

aaa

aaa

det

AA

Are composed of same elements

Completely Different Mathematical Concept

Determinants

2221

1211

aa

aaA

Defined in a recursive form

2x2 matrix

122122112221

1211det aaaaaa

aaA

Determinants

nnnjnn

inijii

nj

nj

aaaa

aaaa

aaaa

aaaa

21

21

222221

111211

A

ijAMinor

Determinants

ijji

ij AACofactor minor1

n

kikik AaA

1

]det[

DeterminantsDefined in a recursive form

3x3 matrix

3231

222113

3331

232112

3332

232211

det

aa

aaa

aa

aaa

aa

aaa

A

333231

232221

131211

aaa

aaa

aaa

333231

232221

131211

aaa

aaa

aaa

Determinants

3332

232211 aa

aaa

3231

222113

3331

232112 aa

aaa

aa

aaa

3332

2322

aa

aaMinor a11

333231

232221

131211

aaa

aaa

aaa

Determinants

3331

2321

aa

aaMinor a12

3332

232211 aa

aaa

3331

232112 aa

aaa

3231

222113 aa

aaa

333231

232221

131211

aaa

aaa

aaa

Determinants

3231

2221

aa

aaMinor a13

3332

232211 aa

aaa

3331

232112 aa

aaa

3231

222113 aa

aaa

DeterminantsProperties1) If two rows or two columns of matrix [A] are equal then det[A]=0

2) Interchanging any two rows or columns will change the sign of the det

3) If a row or a column of a matrix is {0} then det[A]=0

4)

5) If we multiply any row or column by a scalar s then

6) If any row or column is replaced by a linear combination of any of the other rows or columns the value of det[A] remains unchanged

AsAs n

AsA

Operations - Inverse

[A] [A]-1

[A] [A]-1=[I]

If [A]-1 does not exist[A] is singular

Operations - Inverse

Calculation of [A]-1

AadjA

A11

TijAAadj

Solution of Linear Equations

23610

3112

835

z

y

x

24

6

10

bxA

bAxAA 11

bAxI 1 bAx 1

Numerical Solution of Linear Equations

Solution of Linear Equations

9835 zyx

7310 zyx

10500 zyx

Consider the system

Solution of Linear Equations

9835 zyx

730 zyx

10500 zyx

25

10 z

Solution of Linear Equations

9835 zyx

730 zyx

2z

7230 yx

167 y

Solution of Linear Equations9835 zyx

2z

1y

928135 x

25

1639

x 2x

Solution of Linear Equations

10

7

9

500

310

835

z

y

x

Express In Matrix Form

Upper Triangular

What is the characteristic?

Solution by Back Substitution

Solution of Linear EquationsObjective

Can we express any system of equations in a form

nnnn

n

n

n

b

b

b

b

x

x

x

x

a

aa

aaa

aaaa

3

2

1

3

2

1

333

22322

1131211

000

00

0

0

BackgroundConsider

1035 yx(Eq 1)

5810 yx(Eq 2)

Solution

5.7

5.6

y

x

20610 yx2*(Eq 1)

5810 yx(Eq 2)

Solution

5.7

5.6

y

x!!!!!!

Scaling Does Not Change the SolutionScaling Does Not Change the Solution

BackgroundConsider

20610 yx(Eq 1)

152 y(Eq 2)-(Eq 1)

Solution

5.7

5.6

y

x!!!!!!

20610 yx(Eq 1)

5810 yx(Eq 2)

Solution

5.7

5.6

y

x

Operations Do Not Change the SolutionOperations Do Not Change the Solution

Gauss Elimination

10835 zyx

2423610 zyx

6312 zyx

Example

Forward Elimination

Gauss Elimination

10835 zyx

24z23y6x10

zyx 835

5

1210

5

12

6312 zyx

-

305

81

5

310 zyx 302.162.60 zyx

Gauss Elimination

10835 zyx

24z23y6x10

6312 zyx 302.162.60 zyx

Substitute 2nd eq with new

Gauss Elimination

10835 zyx

24z23y6x10

302.162.60 zyx

zyx 835

5

1010

5

10-

439120 zyx

Gauss Elimination

10835 zyx

24z23y6x10

302.162.60 zyx

Substitute 3rd eq with new

439120 zyx

Gauss Elimination

10835 zyx

302.162.60 zyx

439120 zyx

zy 2.162.6

2.6

12 30

2.6

12-

064.62645.700 zyx

Gauss Elimination

10835 zyx

30970 zyx

Substitute 3rd eq with new

439120 zyx 064.62645.700 zyx

Gauss Elimination

064.62

30

10

645.700

2.162.60

835

z

y

x

Gauss Elimination

118.8645.7/064.62 z

0502.26

2.6

118.82.1630

y

6413.0

5

118.880502.26310

x

064.62

30

10

645.700

2.162.60

835

z

y

x

Gauss Elimination – Potential Problem

10830 zyx

2423610 zyx

6312 zyx

Forward Elimination

Gauss Elimination – Potential Problem

10830 zyx

2423610 zyx

6312 zyx

0

12 Division By Zero!!Operation Failed

Gauss Elimination – Potential Problem

10830 zyx

2423610 zyx

6312 zyx

12

0OK!!

Gauss Elimination – Potential Problem

10830 zyx

2423610 zyx

6312 zyx

Pivoting

6312 zyx

10830 zyx

Partial Pivoting

nn

nnnnn

lll

n

n

n

b

b

b

b

x

x

x

x

aaaa

aaaa

aaaaaaaa

aaaa

3

2

1

3

2

1

321

ln321

3333231

2232221

1131211

a32>a22

al2>a22

NO

YES

Partial Pivoting

nn

nnnnn

n

n

lll

n

b

b

b

b

x

x

x

x

aaaa

aaaa

aaaaaaaa

aaaa

3

2

1

3

2

1

321

2232221

3333231

ln321

1131211

Full Pivoting

• In addition to row swaping

• Search columns for max elements

• Swap Columns

• Change the order of xi

• Most cases not necessary

EXAMPLE

4.71

3.19

85.7

102.03.0

3.071.0

2.01.03

3

2

1

x

x

x

Eliminate Column 1

3

1.0

PIVOTS

4.71

3.19

85.7

102.03.0

3.071.0

2.01.03

3

3.0

1,11

11 ia

apivot i

i

njapivotaa jiijij ,,2,1,11

Eliminate Column 1

6150.70

5617.19

85.7

0200.1019000.00

29333.000333.70

2.01.03

Eliminate Column 2

00333.7

19000.0

PIVOTS

6150.70

5617.19

85.7

0200.1019000.00

29333.000333.70

2.01.03

2,22

22 ia

apivot i

i

njapivotaa jiijij ,,2,1,22

Eliminate Column 2

0843.70

5617.19

85.7

01200.1000

29333.000333.70

2.01.03

UpperTriangular

Matrix[ U ]

ModifiedRHS

{ b }

01200.1000

29333.000333.70

2.01.03

LU DecompositionPIVOTS

Column 1PIVOTS

Column 2

03333.0

1.0 02713.0

LU Decomposition

As many as, and in the location of, zeros

UpperTriangular

MatrixU

01200.1000

29333.000333.70

2.01.03

LU DecompositionPIVOTS

Column 1

PIVOTSColumn 2

LowerTriangular

Matrix

1

1

1

0

0

0

L

03333.0

1.0 02713.0

LU Decomposition

102713.01.0

0103333.0

001

=

This is the original matrix!!!!!!!!!!

01200.1000

29333.000333.70

2.01.03

102.03.0

3.071.0

2.01.03

LU Decomposition

4.71

3.19

85.7

102713.01.0

0103333.0

001

3

2

1

y

y

y

4.71

3.19

85.7

102.03.0

3.071.0

2.01.03

3

2

1

x

x

x

[ L ] { y } { b }

[ A ] { x } { b }

LU Decomposition

4.71

3.19

85.7

102713.01.0

0103333.0

001

3

2

1

y

y

y

L y b

85.71 y

5617.190333.03.19 12 yy

0843.70)02713.0(1.04.71 213 yyy

LU Decomposition85.71 y

5617.190333.03.19 12 yy

0843.70)02713.0(1.04.71 213 yyy

0843.70

5617.19

85.7

01200.1000

29333.000333.70

2.01.03

ModifiedRHS

{ b }

LU Decomposition

• Ax=b

• A=LU - LU Decomposition

• Ly=b- Solve for y

• Ux=y - Solve for x

Matrix Inversion

4.71

3.19

85.7

102.03.0

3.071.0

2.01.03

3

2

1

x

x

x

bxA

Matrix Inversion

[A] [A]-1

[A] [A]-1=[I]

If [A]-1 does not exist[A] is singular

Matrix Inversion

b xA bxA 1A 1A

I

Matrix Inversion

bAx 1

Solution

Matrix Inversion

[A] [A]-1=[I]

100

010

001

aaa

aaa

aaa

aaa

aaa

aaa

nnn2n1

2n2221

1n1211

nnn2n1

2n2221

1n1211

Matrix Inversion

100

010

001

aaa

aaa

aaa

aaa

aaa

aaa

nnn2n1

2n2221

1n1211

nnn2n1

2n2221

1n1211

Matrix Inversion

100

010

001

aaa

aaa

aaa

aaa

aaa

aaa

nnn2n1

2n2221

1n1211

nnn2n1

2n2221

1n1211

Matrix Inversion

100

010

001

aaa

aaa

aaa

aaa

aaa

aaa

nnn2n1

2n2221

1n1211

nnn2n1

2n2221

1n1211

Matrix Inversion

• To calculate the invert of a nxn matrix solve n times :

nj

2j

1j

nj

2j

1j

nnn2n1

2n2221

1n1211

a

a

a

aaa

aaa

aaa

nj ,,2,1

otherwise

ji if

0

1ij

Matrix Inversion

• For example in order to calculate the inverse of:

102.03.0

3.071.0

2.01.03

Matrix Inversion

• First Column of Inverse is solution of

0

0

1

a

a

a

102.03.0

3.071.0

2.01.03

31

21

11

Matrix Inversion

0

1

0

a

a

a

102.03.0

3.071.0

2.01.03

32

22

12

• Second Column of Inverse is solution of

Matrix Inversion

• Third Column of Inverse is solution of:

1

0

0

a

a

a

102.03.0

3.071.0

2.01.03

33

23

13

Use LU Decomposition

102713.01.0

0103333.0

001

01200.1000

29333.000333.70

2.01.03

102.03.0

3.071.0

2.01.03

A

Use LU Decomposition – 1st column

• Forward SUBSTITUTION

0

0

1

y

y

y

102713.01.0

0103333.0

001

31

21

11

111 y

03333.00333.00 1121 yy

1009.002713.01.00 211131 yyy

Use LU Decomposition – 1st column

• Back SUBSTITUTION

1009.0

0333.0

1

a

a

a

01200.1000

29333.000333.70

2.01.03

31

21

11

010078.0012.10/1009.0a31

00518.000333.7/a2933.00333.0a 3121

332489.03/a2.0a1.01a 312111

Use LU Decomposition – 2nd Column

• Forward SUBSTITUTION

0

1

0

y

y

y

102713.01.0

0103333.0

001

32

22

12

012 y

122 y

02713.002713.01.00 221232 yyy

Use LU Decomposition – 2nd Column

• Back SUBSTITUTION

02713.0

1

0

a

a

a

01200.1000

29333.000333.70

2.01.03

32

22

12

002709.0012.10/02713.0a32

1429.000333.7/a2933.01a 3222

004944.03/a2.0a1.00a 322212

Use LU Decomposition – 3rd Column

• Forward SUBSTITUTION

1

0

0

y

y

y

102713.01.0

0103333.0

001

33

23

13

013 y

023 y

102713.01.01 231333 yyy

Use LU Decomposition – 3rd Column

• Back SUBSTITUTION

1

0

0

a

a

a

01200.1000

29333.000333.70

2.01.03

33

23

13

09988.0012.10/1a33

004183.000333.7/a2933.00a 3323

006798.03/a2.0a1.00a 332313

Result

102.03.0

3.071.0

2.01.03

A

09988.000271.001008.0

004183.0142903.000518.0

006798.0004944.0332489.0

A 1

Test It

09988.000271.001008.0

004183.0142903.000518.0

006798.0004944.0332489.0

102.03.0

3.071.0

2.01.03

11046.30

1047.31106736.8

0108.11

18

1818

18

Iterative Methods

Recall Techniques for Root finding of Single Equations

Initial Guess

New Estimate

Error Calculation

Repeat until Convergence

Gauss Seidel

3

2

1

3

2

1

333231

232221

131211

b

b

b

x

x

x

aaa

aaa

aaa

11

31321211 a

xaxabx

22

32312122 a

xaxabx

33

23213133 a

xaxabx

Gauss Seidel

11

1

11

1312111

00

a

b

a

aabx

22

23112121

2

0

a

axabx

33

1232

113131

3 a

xaxabx

First Iteration: 0,0,0 321 xxx

Better Estimate

Better Estimate

Better Estimate

Gauss Seidel

11

1313

121212

1 a

xaxabx

22

1323

212122

2 a

xaxabx

33

2232

213132

3 a

xaxabx

Second Iteration: 13

12

11 ,, xxx

Better Estimate

Better Estimate

Better Estimate

Gauss SeidelIteration Error:

%1001

, ji

ji

ji

ia x

xx

s

Convergence Criterion:

n

jij

ijii aa1

nnn2n1

2n2221

1n1211

aaa

aaa

aaa

Jacobi Iteration

3

2

1

3

2

1

333231

232221

131211

b

b

b

x

x

x

aaa

aaa

aaa

11

31321211 a

xaxabx

22

32312122 a

xaxabx

33

23213133 a

xaxabx

Jacobi Iteration

11

1

11

1312111

00

a

b

a

aabx

22

2321212

00

a

aabx

33

3231313

00

a

aabx

First Iteration: 0,0,0 321 xxx

Better Estimate

Better Estimate

Better Estimate

Jacobi Iteration

11

1313

121212

1 a

xaxabx

22

1323

112122

2 a

xaxabx

33

1232

113132

3 a

xaxabx

Second Iteration: 13

12

11 ,, xxx

Better Estimate

Better Estimate

Better Estimate

Jacobi Iteration

Iteration Error:

%1001

, ji

ji

ji

ia x

xx

s