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View File - University of Engineering and Technology, Taxila

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that is, N zeros at the top <strong>and</strong> bottom <strong>of</strong> 1;2½hŠ ¼64a Na Nþ1a Nþ2.a 0a 1..a N375ð3:32bÞ<strong>and</strong>23z c ð0Þ z c ð TÞ ... z c ð 2NTÞ z c ðTÞ z c ð0Þ ... z c ½ð 2N þ 1ÞTŠz c ¼ . . ... . 6745z c ð2NTÞ z c ½ð2N 1ÞTŠ ... z c ð0Þð3:32cÞThe preceding procedure describes the zero forcing technique. Since ½z eq Š isspecified by the zero forcing condition, all that is required is to find ½z c Š 1 ; thatis, the inverse <strong>of</strong> ½z c Š. The desired coefficient matrix ½hŠ is then the middlecolumn <strong>of</strong> ½z c Š1 , which follows by multiplying ½z c Š1 <strong>and</strong> ½z eq Š.As a refresher, the inverse <strong>of</strong> a nonsingular matrix A is defined asA 1 ¼ A adjjAjwhere A adj denotes the adjoint matrix <strong>of</strong> A, <strong>and</strong> jAj denotes the determinant <strong>of</strong>A. For example, if A is a square matrix, its adjoint is the transpose <strong>of</strong> thematrix obtained from A by replacing each element <strong>of</strong> A by its c<strong>of</strong>actor. Thisdefinition becomes clearer with an illustration.Example 3.1: Find the inverse <strong>of</strong> matrix A defined by231 2 0A ¼ 4 3 1 2 5 ð3:33aÞ1 0 3Copyright © 2002 by Marcel Dekker, Inc. All Rights Reserved.

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