ECIV 301 Programming & Graphics Numerical Methods for Engineers Lecture 14 Elimination Methods

Objectives Introduction to Matrix Algebra Express System of Equations in Matrix Form Introduce Methods for Solving Systems of Equations Advantages and Disadvantages of each Method

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

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

Last Time Matrix Algebra 3 rd Row 2 nd Column

Last Time Matrix Algebra 1 Row, m Columns Row Vector

Last Time Matrix Algebra n Rows, 1 Column Column Vector

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

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

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

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

Last Time Matrix Algebra Upper Triangular Special Types of Square Matrices

Last Time Matrix Algebra Lower Triangular Special Types of Square Matrices

Last Time Matrix Algebra Banded Special Types of Square Matrices

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

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

Last Time Multiplication by Scalar

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

Last Time Matrix Multiplication

Last Time Operations - Transpose

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

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

Equations in Matrix Form Consider

Linear Equations in Matrix Form

# Equations = # Unknowns = n Square Matrix n x n

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