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ECIV 520 Structural Analysis II Review of Matrix Algebra

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Linear Equations in Matrix Form

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

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

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Matrix Algebra 3 rd Row 2 nd Column

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Matrix Algebra 1 Row, m Columns Row Vector

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Matrix Algebra n Rows, 1 Column Column Vector

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Matrix Algebra If n = m Square Matrix e.g. n=m=5 Main Diagonal

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

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

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

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Matrix Algebra Upper Triangular Special Types of Square Matrices

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Matrix Algebra Lower Triangular Special Types of Square Matrices

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Matrix Algebra Banded Special Types of Square Matrices

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

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

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

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Multiplication by Scalar

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

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Matrix Multiplication

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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]

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Operations - Transpose

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

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Determinants Are composed of same elements Completely Different Mathematical Concept

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Determinants Defined in a recursive form 2x2 matrix

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Determinants

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Defined in a recursive form 3x3 matrix

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Determinants Minor a 11

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Determinants Minor a 12

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Determinants Minor a 13

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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

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

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Operations - Inverse Calculation of [A] -1

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Solution of Linear Equations

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Numerical Solution of Linear Equations

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Solution of Linear Equations Consider the system

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Solution of Linear Equations

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

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

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

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

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Gauss Elimination Example Forward Elimination

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Gauss Elimination -

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Substitute 2 nd eq with new

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Gauss Elimination -

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Substitute 3rd eq with new

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Gauss Elimination -

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Substitute 3rd eq with new

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Gauss Elimination

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Gauss Elimination – Potential Problem Forward Elimination

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Gauss Elimination – Potential Problem Division By Zero!! Operation Failed

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Gauss Elimination – Potential Problem OK!!

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Gauss Elimination – Potential Problem Pivoting

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Partial Pivoting a 32 >a 22 a l2 >a 22 NO YES

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Partial Pivoting

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Full Pivoting In addition to row swaping Search columns for max elements Swap Columns Change the order of x i Most cases not necessary

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EXAMPLE

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Eliminate Column 1 PIVOTS

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Eliminate Column 1

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Eliminate Column 2 PIVOTS

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Eliminate Column 2 Upper Triangular Matrix [ U ] Modified RHS { b }

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LU Decomposition PIVOTS Column 1 PIVOTS Column 2

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

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

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

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

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LU Decomposition Lyb

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Modified RHS { b }

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

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Matrix Inversion

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

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Matrix Inversion

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Solution

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

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Matrix Inversion

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

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Matrix Inversion For example in order to calculate the inverse of:

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Matrix Inversion First Column of Inverse is solution of

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Matrix Inversion Second Column of Inverse is solution of

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Matrix Inversion Third Column of Inverse is solution of:

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Use LU Decomposition

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Use LU Decomposition – 1 st column Forward SUBSTITUTION

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Use LU Decomposition – 1 st column Back SUBSTITUTION

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Use LU Decomposition – 2 nd Column Forward SUBSTITUTION

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Use LU Decomposition – 2 nd Column Back SUBSTITUTION

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Use LU Decomposition – 3 rd Column Forward SUBSTITUTION

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Use LU Decomposition – 3 rd Column Back SUBSTITUTION

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Result

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Test It

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Iterative Methods Recall Techniques for Root finding of Single Equations Initial Guess New Estimate Error Calculation Repeat until Convergence

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Gauss Seidel

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First Iteration: Better Estimate

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Gauss Seidel Second Iteration: Better Estimate

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Gauss Seidel Iteration Error: Convergence Criterion:

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Jacobi Iteration

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First Iteration: Better Estimate

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Jacobi Iteration Second Iteration: Better Estimate

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Jacobi Iteration Iteration Error:

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