ECIV 301

Programming & Graphics

Numerical Methods for Engineers

Lecture 12

System of Linear Equations

Objectives

• Introduction to Matrix Algebra

• Express System of Equations in Matrix Form

• Introduce Methods for Solving Systems of Equations

• Advantages and Disadvantages of each Method

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

Example

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 - Inverse

[A] [A]-1

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

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

Operations - Trace

5554535251

4544434241

3534333231

2524232221

1514131211

aaaaa

aaaaa

aaaaa

aaaaa

aaaaa

A

Square Matrix

tr[A] = tr[A] = aaiiii

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

Homework

Problems 9.1, 9.2, 9.3

Due Date: Oct 6