eciv 301
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
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
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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
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aaaaa
aaaaa
aaaaa
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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
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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
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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