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: