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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 , ir have been replaced by −eT1 , −eT2 , .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 , ir have been replaced by eT1 , eT2 , .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 meyer@ncsu.edu 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...
View Full Document

Ask a homework question - tutors are online