Download presentation

Presentation is loading. Please wait.

1
ECIV 520 Structural Analysis II Review of Matrix Algebra

2
Linear Equations in Matrix Form

7
Matrix Algebra Rectangular Array of Elements Represented by a single symbol [A]

8
Matrix Algebra Row 1 Row 3 Column 2Column m n x m Matrix

9
Matrix Algebra 3 rd Row 2 nd Column

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. n=m=5 Main Diagonal

13
Matrix Algebra Special Types of Square Matrices Symmetric: a ij = a ji

14
Matrix Algebra Diagonal: a ij = 0, i j Special Types of Square Matrices

15
Matrix Algebra Identity: a ii =1.0 a ij = 0, i j Special Types of Square Matrices

16
Matrix Algebra Upper Triangular Special Types of Square Matrices

17
Matrix Algebra Lower Triangular Special Types of Square Matrices

18
Matrix Algebra Banded Special Types of Square Matrices

19
Matrix Operating Rules - Equality [A] mxn =[B] pxq n=pm=qa ij =b ij

20
Matrix Operating Rules - Addition [C] mxn = [A] mxn +[B] pxq n=p m=q c ij = a ij +b ij

21
Matrix Operating Rules - Addition Properties [A]+[B] = [B]+[A] [A]+([B]+[C]) = ([A]+[B])+[C]

22
Multiplication by Scalar

23
Matrix Multiplication [A] n x m. [B] p x q = [C] n x q m=p

24
Matrix Multiplication

26
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]

27
Operations - Transpose

28
Operations - Trace Square Matrix tr[A] = a ii

29
Determinants Are composed of same elements Completely Different Mathematical Concept

30
Determinants Defined in a recursive form 2x2 matrix

31
Determinants

33
Defined in a recursive form 3x3 matrix

34
Determinants Minor a 11

35
Determinants Minor a 12

36
Determinants Minor a 13

37
Determinants Properties 1)If two rows or two columns of matrix [A] are equal then det[A]=0 2)Interchanging any two rows or columns will change the sign of the det 3)If a row or a column of a matrix is {0} then det[A]=0 4) 5)If we multiply any row or column by a scalar s then 6) If any row or column is replaced by a linear combination of any of the other rows or columns the value of det[A] remains unchanged

38
Operations - Inverse [A][A] -1 [A] [A] -1 =[I] 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

46
Express In Matrix Form Upper Triangular What is the characteristic? Solution by Back Substitution

47
Solution of Linear Equations Objective Can we express any system of equations in a form 0

48
Background Consider (Eq 1) (Eq 2) Solution 2*(Eq 1) (Eq 2) Solution !!!!!! Scaling Does Not Change the Solution

49
Background Consider (Eq 1) (Eq 2)-(Eq 1) Solution !!!!!! (Eq 1) (Eq 2) Solution Operations Do Not Change the Solution

50
Gauss Elimination Example Forward Elimination

51
Gauss Elimination -

52
Substitute 2 nd eq with new

53
Gauss Elimination -

54
Substitute 3rd eq with new

55
Gauss Elimination -

56
Substitute 3rd eq with new

57
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 a 32 >a 22 a l2 >a 22 NO YES

64
Partial Pivoting

65
Full Pivoting In addition to row swaping Search columns for max elements Swap Columns Change the order of x i 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 [ U ] Modified RHS { b }

71
LU Decomposition PIVOTS Column 1 PIVOTS Column 2

72
LU Decomposition As many as, and in the location of, zeros Upper Triangular Matrix U

73
LU Decomposition PIVOTS Column 1 PIVOTS Column 2 Lower Triangular Matrix L

74
LU Decomposition = This is the original matrix!!!!!!!!!!

75
LU Decomposition [ L ]{ y }{ b } [ A ]{ x }{ b }

76
LU Decomposition Lyb

77
Modified RHS { b }

78
LU Decomposition Ax=b A=LU -LU Decomposition Ly=b- Solve for y Ux=y- Solve for x

79
Matrix Inversion

80
[A][A] -1 [A] [A] -1 =[I] If [A] -1 does not exist [A] is singular

81
Matrix Inversion

82
Solution

83
Matrix Inversion [A] [A] -1 =[I]

84
Matrix Inversion

87
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 – 1 st column Forward SUBSTITUTION

94
Use LU Decomposition – 1 st column Back SUBSTITUTION

95
Use LU Decomposition – 2 nd Column Forward SUBSTITUTION

96
Use LU Decomposition – 2 nd Column Back SUBSTITUTION

97
Use LU Decomposition – 3 rd Column Forward SUBSTITUTION

98
Use LU Decomposition – 3 rd 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
First Iteration: Better Estimate

104
Gauss Seidel Second Iteration: Better Estimate

105
Gauss Seidel Iteration Error: Convergence Criterion:

106
Jacobi Iteration

107
First Iteration: Better Estimate

108
Jacobi Iteration Second Iteration: Better Estimate

109
Jacobi Iteration Iteration Error:

Similar presentations

© 2019 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google