ECIV 301 Programming & Graphics Numerical Methods for Engineers Lecture 12 System of Linear Equations

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

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

Example

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

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

Linear Equations in Matrix Form

Homework Problems 9.1, 9.2, 9.3 Due Date: Oct 6