ECIV 520

1
ECIV 520

• Structural Analysis II
• Review of Matrix Algebra

2
Linear Equations in Matrix Form
3
Linear Equations in Matrix Form
4
Linear Equations in Matrix Form
5
Linear Equations in Matrix Form
6
(No Transcript)
7
Matrix Algebra
Rectangular Array of Elements Represented by a
single symbol A
8
Matrix Algebra
n x m Matrix
9
Matrix Algebra
10
Matrix Algebra
1 Row, m Columns
Row Vector
11
Matrix Algebra
n Rows, 1 Column
Column Vector
12
Matrix Algebra
If n m Square Matrix
e.g. nm5
13
Matrix Algebra
Special Types of Square Matrices
Symmetric aij aji
14
Matrix Algebra
Special Types of Square Matrices
Diagonal aij 0, i?j
15
Matrix Algebra
Special Types of Square Matrices
Identity aii1.0 aij 0, i?j
16
Matrix Algebra
Special Types of Square Matrices
Upper Triangular
17
Matrix Algebra
Special Types of Square Matrices
Lower Triangular
18
Matrix Algebra
Special Types of Square Matrices
Banded
19
Matrix Operating Rules - Equality
AmxnBpxq
np
mq
aijbij
20
Matrix Operating Rules - Addition
Cmxn AmxnBpxq
np
cij aijbij
mq
21
Matrix Operating Rules - Addition
Properties
AB BA
A(BC) (AB)C
22
Multiplication by Scalar
23
Matrix Multiplication
A n x m . B p x q C n x q
24
Matrix Multiplication
25
Matrix Multiplication
26
Matrix Multiplication - Properties
If dimensions suitable
Associative A(BC) (AB)C
Distributive A(BC) ABA C
Attention AB ? BA
27
Operations - Transpose
28
Operations - Trace
Square Matrix
trA Saii
29
Determinants
Are composed of same elements
Completely Different Mathematical Concept
30
Determinants
Defined in a recursive form
2x2 matrix
31
Determinants
32
Determinants
33
Determinants
Defined in a recursive form
3x3 matrix
34
Determinants
Minor a11
35
Determinants
Minor a12
36
Determinants
Minor a13
37
Determinants

• Properties
• If two rows or two columns of matrix A are
  equal then detA0
• Interchanging any two rows or columns will change
  the sign of the det
• If a row or a column of a matrix is 0 then
  detA0
• If we multiply any row or column by a scalar s
  then
• If any row or column is replaced by a linear
  combination of any of the other rows or columns
  the value of detA remains unchanged

38
Operations - Inverse
A
A-1
A A-1I
If A-1 does not exist A is singular
39
Operations - Inverse
Calculation of A-1
40
Solution of Linear Equations
41
Numerical Solution of Linear Equations
42
Solution of Linear Equations
Consider the system
43
Solution of Linear Equations
44
Solution of Linear Equations
45
Solution of Linear Equations
46
Solution of Linear Equations
Express In Matrix Form
What is the characteristic?
Upper Triangular
Solution by Back Substitution
47
Solution of Linear Equations
Objective
Can we express any system of equations in a form
48
Background
Consider
(Eq 1)
2(Eq 1)
(Eq 2)
(Eq 2)
Solution
Solution
!!!!!!
Scaling Does Not Change the Solution
49
Background
Consider
(Eq 1)
(Eq 1)
(Eq 2)-(Eq 1)
(Eq 2)
Solution
Solution
!!!!!!
Operations Do Not Change the Solution
50
Gauss Elimination
Example
Forward Elimination
51
Gauss Elimination
-
52
Gauss Elimination
Substitute 2nd eq with new
53
Gauss Elimination
-
54
Gauss Elimination
Substitute 3rd eq with new
55
Gauss Elimination
-
56
Gauss Elimination
Substitute 3rd eq with new
57
Gauss Elimination
58
Gauss Elimination
59
Gauss Elimination Potential Problem
Forward Elimination
60
Gauss Elimination Potential Problem
Division By Zero!! Operation Failed
61
Gauss Elimination Potential Problem
OK!!
62
Gauss Elimination Potential Problem
Pivoting
63
Partial Pivoting
NO
YES
64
Partial Pivoting
65
Full Pivoting

• In addition to row swaping
• Search columns for max elements
• Swap Columns
• Change the order of xi
• Most cases not necessary

66
EXAMPLE
67
Eliminate Column 1
PIVOTS
68
Eliminate Column 1
69
Eliminate Column 2
PIVOTS
70
Eliminate Column 2
Upper Triangular Matrix
Modified RHS
U
71
LU Decomposition
PIVOTS Column 1
PIVOTS Column 2
72
LU Decomposition
Upper Triangular Matrix
U
As many as, and in the location of, zeros
73
LU Decomposition
PIVOTS Column 2
PIVOTS Column 1
Lower Triangular Matrix
L
74
LU Decomposition

This is the original matrix!!!!!!!!!!
75
LU Decomposition
A
x
b
L
y
b
76
LU Decomposition
L
y
b
77
LU Decomposition
Modified RHS
78
LU Decomposition

• Axb
• ALU - LU Decomposition
• Lyb - Solve for y
• Uxy - Solve for x

79
Matrix Inversion
80
Matrix Inversion
A
A-1
A A-1I
If A-1 does not exist A is singular
81
Matrix Inversion
82
Matrix Inversion
Solution
83
Matrix Inversion
A A-1I
84
Matrix Inversion
85
Matrix Inversion
86
Matrix Inversion
87
Matrix Inversion

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

88
Matrix Inversion

• For example in order to calculate the inverse of

89
Matrix Inversion

• First Column of Inverse is solution of

90
Matrix Inversion

• Second Column of Inverse is solution of

91
Matrix Inversion

• Third Column of Inverse is solution of

92
Use LU Decomposition
93
Use LU Decomposition 1st column

• Forward SUBSTITUTION

94
Use LU Decomposition 1st column

• Back SUBSTITUTION

95
Use LU Decomposition 2nd Column

• Forward SUBSTITUTION

96
Use LU Decomposition 2nd Column

• Back SUBSTITUTION

97
Use LU Decomposition 3rd Column

• Forward SUBSTITUTION

98
Use LU Decomposition 3rd Column

• Back SUBSTITUTION

99
Result
100
Test It
101
Iterative Methods
Recall Techniques for Root finding of Single
Equations
Initial Guess New Estimate Error
Calculation Repeat until Convergence
102
Gauss Seidel
103
Gauss Seidel
First Iteration
104
Gauss Seidel
Second Iteration
105
Gauss Seidel
Iteration Error
Convergence Criterion
106
Jacobi Iteration
107
Jacobi Iteration
First Iteration
108
Jacobi Iteration
Second Iteration
109
Jacobi Iteration
Iteration Error