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

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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

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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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Example

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

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

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Linear Equations in Matrix Form

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

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