Chapter 8 Linear Algebraic Equations and Matrices

Chapter 8Chapter 8

Linear Algebraic Equations Linear Algebraic Equations and Matricesand Matrices

Three individuals connected by bungee cords

Free-body diagramsFree-body diagrams

Newton’s Newton’s second lawsecond law

0xxkgm

0xxkxxkgm

0xkxxkgm

2333

1222332

111221

)(

)()(

)(

gmxkxk

gmxkxkkxk

gmxkxkk

33323

23323212

122121

)(

)(

Rearrange the equations

[K] {x} = {b}

Newton’s second law – equation of motionNewton’s second law – equation of motion

Kirchhoff’s current and voltage rules

Mass-spring system

(similar to bungee jumpers)

Resistor circuits

Solved single equations previously

Now consider more than one variable and more than one equation

0xf

0x,...,x,xf

0x,...,x,xf0x,...,x,xf

n21n

n212

n211

Linear Algebraic EquationsLinear Algebraic Equations

Linear equations and constant coefficients

aij and bi are constants

nnnn22n11n

2nn2222121

1nn1212111

bxa...xaxa

bxa...xaxabxa...xaxa

Linear SystemsLinear Systems

nnnn22n11n

2nn2222121

1nn1212111

RFa...FaFa

RFa...FaFa

RFa...FaFa

Forces on a TrussForces on a TrussMost obvious example in Civil Engineeringtrusses: force balance at joints

F1

F2F3

R

}{}]{[][]][[ bxA or bxA

n

2

1

n

2

1

nn2n1n

n22221

n11211

b

b

b

x

x

x

aaa

aaa

aaa

Mathematical backgroundMathematical background

It is convenient to write system of equations in matrix-vector form

mn4m3m2m1m

n444434241

n334333231

n224232221

n114131211

aaaaa

aaaaa

aaaaa

aaaaa

aaaaa

A

Matrix NotationsMatrix Notations

Column 4

Row 3

(second index)

(first index)

Scalars, Vectors, MatricesScalars, Vectors, Matrices

MATLAB treat variables as “matrices”

Matrix (m n) - a set of numbers arranged in rows (m) and columns (n)

Scalar : 1 1 matrixRow Vector : 1 n matrix ( [b] or b )Column Vector : m 1 matrix ( [c] or {c} )

227150

592342

5231

D

217

32

025

bbc

21732025bb 275a

T

..

..][

.

.

.

][}{][

...][.][

Square matrix, m = nParticularly important when solving simultaneous

equations in engineering applications

aii – principle or main diagonal

44434241

34333231

24232221

14131211

aaaa

aaaa

aaaa

aaaa

A][

Square MatrixSquare Matrix

Transpose

In MATLAB, transpose is A

Trace is sum of diagonal elementsIn MATLAB, trace(A)

44434241

34333231

24232221

14131211

aaaa

aaaa

aaaa

aaaa

A][

44342414

43332313

42322212

41312111

T

aaaa

aaaa

aaaa

aaaa

A][

Matrix OperationsMatrix Operations

Matrix TransposeMatrix Transpose

4231234

2

1

3

324yx

639

426

8412

213

3

2

4

yx

2

1

3

y' ;

3

2

4

x

213y ;324x

)())(())((*

*

43016

31254

02381

15822

64125

B

jigd

ihfc

gfeb

dcba

A

][

][

• symmetric matrices

Special MatricesSpecial Matrices

aij = aji

[A]T = [A]

44

33

22

11

a

a

a

a

A][

• Diagonal matrix • Identity matrix

1

1

1

1

I ][

Special MatricesSpecial Matrices

[A][I] = [I][A] = [A]

Banded matrix – all elements are zero, with the exception of a band centered on the main diagonal

Special MatricesSpecial Matrices

4443

343332

232221

1211

aa

aaa

aaa

aa

A][Tridiagonal – three

non-zero bands

• lower triangular

44434241

333231

2221

11

aaaa

aaa

aa

a

A][

• upper triangular

44

3433

242322

14131211

a

aa

aaa

aaaa

A][

Special MatricesSpecial Matrices

Matrix Operation RulesMatrix Operation Rules

Matrix identity

[A] = [B] if and only if aij = bij for all i and j

Matrix Addition and Subtraction

[C] = [A] + [B] Cij = Aij + Bij

[C] = [A] [B] Cij = Aij Bij

Addition and SubtractionAddition and Subtraction

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

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

Multiplication of Matrix by a ScalarMultiplication of Matrix by a Scalar

6132

0725

3124

4231

aaaa

aaaa

aaaa

aaaa

A

44434241

34333231

24232221

14131211

][

3051510

0351025

1551020

2010155

gagagaga

gagagaga

gagagaga

gagagaga

gAAgB

44434241

34333231

24232221

14131211

][][][

g = 5

Visual depiction of how the rows and columns line up in matrix multiplication

Matrix MultiplicationMatrix Multiplication

Matrix MultiplicationsMatrix Multiplications

Recall how matrix multiplication works

lficleibldia

kfhckehbkdha

jfgcjegbjdga

fed

cba

li

kh

jg

flekdjfiehdg

clbkajcibhag

li

kh

jg

fed

cba

[A]*[B] [B]*[A]

Matrix multiplication can be performed only if the inner dimensions are equal

Matrix MultiplicationMatrix Multiplication

Interior dimensions have to be equal

For a vector

We will be using square matrices

kikjji CBA

ijji bxA

nnnn bxA

Matrix MultiplicationMatrix Multiplication

MATLABMATLAB

In Fortran, the matrix multiplication have to be done by Do Loops

In MATLAB, it is automatic

A*B = C Note no period ‘.’ (not element-by-element operation)

For vectors

A*x = b

Matrix MultiplicationMatrix Multiplication

Associative

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

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

([A] + [B] ) [C] = [A] [C] + [B] [C] Not generally commutative

[A] [B] [B] [A]

Matrix InverseMatrix Inverse

Matrix division is undefined However, there is a matrix inverse for

non-singular square matrices

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

Multiplication of a matrix by the inverse is analogous to division

1000

0100

0010

0001

aaaa

aaaa

aaaa

aaaa

A

44434241

34333231

24232221

14131211

44434241

34333231

24232221

14131211

aaaa

aaaa

aaaa

aaaa

A

AugmentationAugmentation

Whatever you do to left-hand-side, do to the right-hand side (useful when solving system of equations)

444434241

334333231

224232221

114131211

baaaa

baaaa

baaaa

baaaa

bA

Augmented MatrixAugmented Matrix

4444343242141

3434333232131

2424323222121

1414313212111

bxaxaxaxa

bxaxaxaxa

bxaxaxaxa

b xaxaxaxa

MATLAB Matrix ManipulationsMATLAB Matrix Manipulations

>> A = [1 5 3 -4; 2 5 6 -1; 3 4 -2 5; -1 3 2 6]

A =

1 5 3 -4

2 5 6 -1

3 4 -2 5

-1 3 2 6

>> A'

ans =

1 2 3 -1

5 5 4 3

3 6 -2 2

-4 -1 5 6

Create a matrix

Matrix transpose

MATLAB Matrix ManipulationsMATLAB Matrix Manipulations

>> x = [5 1 -2 3];

>> y = [2 -1 4 2];

>> z = [3 -2 1 -5];

>> B = [x; y; x; z]

B =

5 1 -2 3

2 -1 4 2

5 1 -2 3

3 -2 1 -5

>> C = A + B

C =

6 6 1 -1

4 4 10 1

8 5 -4 8

2 1 3 1

>> C = C – B ( = A)C =

1 5 3 -4

2 5 6 -1

3 4 -2 5

-1 3 2 6

Matrix Concatenation

Addition and Subtraction

MATLAB Matrix ManipulationsMATLAB Matrix Manipulations

A =

1 5 3 -4

2 5 6 -1

3 4 -2 5

-1 3 2 6

B =

5 1 -2 3

2 -1 4 2

5 1 -2 3

3 -2 1 -5

>> A*B

ans =

18 7 8 42

47 5 3 39

28 -13 19 -14

29 -14 16 -21

>> A.*B

ans =

5 5 -6 -12

4 -5 24 -2

15 4 4 15

-3 -6 2 -30

Matrix multiplication and element-by-element operation

A*B A*B A.*B A.*B

MATLAB Matrix ManipulationsMATLAB Matrix Manipulations

>> D = [2 4 3 1; 3 -5 1 2; 1 -1 3 2]

D =

2 4 3 1

3 -5 1 2

1 -1 3 2

>> A*D

??? Error using ==> *

Inner matrix dimensions must agree.

>> D*A

ans =

18 45 26 9

-6 0 -19 10

6 18 -5 24

A =

1 5 3 -4

2 5 6 -1

3 4 -2 5

-1 3 2 6

Inner dimension must agree

A*D D*A

MATLAB Matrix ManipulationsMATLAB Matrix Manipulations

>> A = [1 5 3 -4; 2 5 6 -1; 3 4 -2 5; -1 3 2 6]

>> format short; AI = inv(A)

AI =

-0.2324 0.2520 0.1606 -0.2467

0.2428 -0.1397 0.0457 0.1005

-0.1436 0.2063 -0.0862 0.0104

-0.1123 0.0431 0.0326 0.0718

>> A*AI

ans =

1.0000 0 0 -0.0000

-0.0000 1.0000 0 0.0000

0.0000 0 1.0000 -0.0000

0.0000 0 0.0000 1.0000

Matrix Inverse

MATLAB Matrix ManipulationsMATLAB Matrix Manipulations

>> A = [1 5 3 -4; 2 5 6 -1; 3 4 -2 5; -1 3 2 6]>> I = eye(4)I = 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1>> Aug = [A I]Aug = 1 5 3 -4 1 0 0 0 2 5 6 -1 0 1 0 0 3 4 -2 5 0 0 1 0 -1 3 2 6 0 0 0 1>> [n, m] = size(Aug)n = 4m = 8

Matrix Augmentation

Bungee JumpersBungee Jumpers

gm

gm

gm

x

x

x

kk0

kkkk

0kkk

0xxkgm

0xxkxxkgm

0xkxxkgm

3

2

1

3

2

1

33

3322

221

2333

1222332

111221

)(

)(

)(

)()(

)(

[K] {x} = {b}

Jumper Mass (kg) Spring constant (N/m) Unstretched cord length (m)

Top (1) 60 50 20

Middle (2) 70 100 20

Bottom (3) 80 50 20

>> k1 = 50; k2 = 100; k3 = 50;>> K=[k1+k2 -k2 0;-k2 k2+k3 -k3;0 -k3 k3]K = 150 -100 0 -100 150 -50 0 -50 50>> format short>> g = 9.81; mg = [60; 70; 80]*gmg = 588.6000 686.7000 784.8000

>> x=K\mgx = 41.2020 55.9170 71.6130

>> xi = [20; 40; 60]; >> xf = xi + xxf = 61.2020 95.9170 131.6130

>> x = inv(K)*mgx = 41.2020 55.9170 71.6130

k1 = 50

k2 = 100 stiffer cord

k3 = 50

Final positions of bungee jumpers

1

4

2 3

5

F14

F23F12

F24

F45

H1

F35

F25

V3V1

TRUSSTRUSS

W = 100 kg

Statics: Force BalanceStatics: Force Balance

0FFFF

0FFF

0FFFF

0FFF

0FFF

0FVF

0FFFFF

100FFF

0FFHF

0FVF

4535255x

35255y

4524144x

24144y

35233x

3533y

252423122x

25242y

141211x

1411y

coscos

sinsin

coscos

sinsin

cos

sin

coscos

sinsin

sin

sin

,

,

,

,

,

,

,

,

,

,

Node 1

Node 2

Node 3

Node 4

Node 5

Exampe: Forces in a Simple TrussExampe: Forces in a Simple Truss

0

0

0

0

0

0

0

100

0

0

F

F

F

F

F

F

F

V

H

V

1coscos0000000

0sinsin0000000

100cos0cos0000

000sin0sin0000

0cos00100000

0sin00000100

00coscos101000

00sinsin000000

00000cos1010

00000sin0001

45

35

25

24

23

14

12

3

1

1

function [A, b]=Truss(alpha, beta, gamma, delta)

A = zeros(10,10);A(1,1) = 1; A(1,5) = sin(alpha);A(2,2) = 1; A(2,4) = 1; A(2,5) = cos(alpha);A(3,7) = sin(beta); A(3,8) = sin(gamma);A(4,4) = -1; A(4,6) = 1; A(4,7) = -cos(beta); A(4,8) = cos(gamma);A(5,3)= 1; A(5,9) = sin(gamma);A(6,6) = -1; A(6,9) = -cos(delta);A(7,5) = -sin(alpha); A(7,7)=-sin(beta);A(8,5) = -cos(alpha); A(8,7) = cos(beta); A(8,10)=1;A(9,8) = -sin(gamma); A(9,9) = -sin(delta);A(10,8) = -cos(gamma); A(10,9) = cos(delta); A(10,10) = -1;

b = zeros(10,1); b(3,1)=100;f = A\b

Define Matrices A and b in script file

[A]{ f } = {b} { f } = [A]1 {b}