110 and expand the left hand side to produce n s1 n1

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Unformatted text preview: H Di1 ···ir (λ), ij =ik where Di1 ···ir (λ) is the determinant of the matrix identical to A − λI except that rows i1 , i2 , . . . , ir have been replaced by −eT1 , −eT2 , . . . , −eTr , respectively. i i i It follows that Di1 ···ir (0) = (−1)r det (Ai1 ···ir ), where Ai1 i2 ···ir is identical to A except that rows i1 , i2 , . . . , ir have been replaced by eT1 , eT2 , . . . , eTr , rei i i spectively, and det (Ai1 ···ir ) is the n − r × n − r principal minor obtained by deleting rows and columns i1 , i2 , . . . , ir from A. Consequently, IG R Y P Di1 ···ir (0) = (−1)r p(r) (0) = ij =ik = r! × (−1)r O C det (Ai1 ···ir ) ij =ik (all n − r × n − r principal minors). The factor r! appears because each of the r! permutations of the subscripts on Ai1 ···ir describes the same matrix. Therefore, (7.1.9) says cn−r = (−1)n (r) p (0) = (−1)n−r r! (all n − r × n − r principal minors). To prove (7.1.6), write the characteristic equation for A as (λ − λ1 )(λ − λ2 ) · · · (λ − λn ) = 0, (7.1.10) and expand the left-hand side to produce λn − s1 λn−1 + · · · + (−1)k sk λn−k + · · · + (−1)n sn = 0. (7.1.11) (Using n = 3 or n = 4 in (7.1.10) makes this clear.) Comparing (7.1.11) with (7.1.5) produces the desired conclusion. Statements (7.1.7) and (7.1.8) are obtained from (7.1.5) and (7.1.6) by setting k = 1 and k = n. Copyright c 2000 SIAM Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 496 Chapter 7 Eigenvalues and Eigenvectors http://www.amazon.com/exec/obidos/ASIN/0898714540 Example 7.1.2 Problem: Determine the eigenvalues and eigenvectors of ⎛ ⎞ −3 1 −3 A = ⎝ 20 3 10 ⎠ . 2 −2 4 It is illegal to print, duplicate, or distribute this material Please report violations to [email protected] Solution: Use the principal minors computed in (7.1.4) along with (7.1.5) to obtain the characteristic equation D E λ3 − 4λ2 − 3λ + 18 = 0. A result from elementar...
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This document was uploaded on 03/06/2014 for the course MA 5623 at City University of Hong Kong.

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