eciv 301
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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
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