ECIV 520 Structural Analysis II Review of Matrix Algebra

Linear Equations in Matrix Form

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

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

Matrix Algebra 3 rd Row 2 nd Column

Matrix Algebra 1 Row, m Columns Row Vector

Matrix Algebra n Rows, 1 Column Column Vector

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

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

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

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

Matrix Algebra Upper Triangular Special Types of Square Matrices

Matrix Algebra Lower Triangular Special Types of Square Matrices

Matrix Algebra Banded Special Types of Square Matrices

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

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

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

Multiplication by Scalar

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

Matrix Multiplication

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]

Operations - Transpose

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

Determinants Are composed of same elements Completely Different Mathematical Concept

Determinants Defined in a recursive form 2x2 matrix

Determinants

Defined in a recursive form 3x3 matrix

Determinants Minor a 11

Determinants Minor a 12

Determinants Minor a 13

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

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

Operations - Inverse Calculation of [A] -1

Solution of Linear Equations

Numerical Solution of Linear Equations

Solution of Linear Equations Consider the system

Solution of Linear Equations

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

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

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

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

Gauss Elimination Example Forward Elimination

Gauss Elimination -

Substitute 2 nd eq with new

Gauss Elimination -

Substitute 3rd eq with new

Gauss Elimination -

Substitute 3rd eq with new

Gauss Elimination

Gauss Elimination – Potential Problem Forward Elimination

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

Gauss Elimination – Potential Problem OK!!

Gauss Elimination – Potential Problem Pivoting

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

Partial Pivoting

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

EXAMPLE

Eliminate Column 1 PIVOTS

Eliminate Column 1

Eliminate Column 2 PIVOTS

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

LU Decomposition PIVOTS Column 1 PIVOTS Column 2

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

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

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

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

LU Decomposition Lyb

Modified RHS { b }

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

Matrix Inversion

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

Matrix Inversion

Solution

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

Matrix Inversion

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

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

Matrix Inversion First Column of Inverse is solution of

Matrix Inversion Second Column of Inverse is solution of

Matrix Inversion Third Column of Inverse is solution of:

Use LU Decomposition

Use LU Decomposition – 1 st column Forward SUBSTITUTION

Use LU Decomposition – 1 st column Back SUBSTITUTION

Use LU Decomposition – 2 nd Column Forward SUBSTITUTION

Use LU Decomposition – 2 nd Column Back SUBSTITUTION

Use LU Decomposition – 3 rd Column Forward SUBSTITUTION

Use LU Decomposition – 3 rd Column Back SUBSTITUTION

Result

Test It

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

Gauss Seidel

First Iteration: Better Estimate

Gauss Seidel Second Iteration: Better Estimate

Gauss Seidel Iteration Error: Convergence Criterion:

Jacobi Iteration

First Iteration: Better Estimate

Jacobi Iteration Second Iteration: Better Estimate

Jacobi Iteration Iteration Error: