Ec securehostcomsiamot71html it is illegal to print

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: order n. Verify these facts for the circulant below by computing its eigenvalues and eigenvectors directly: ⎛ ⎞ 1010 ⎜0 1 0 1⎟ C=⎝ ⎠. 1010 0101 P O C 7.2.21. Suppose that (λ, x) and (µ, y∗ ) are right-hand and left-hand eigenpairs for A ∈ n×n —i.e., Ax = λx and y∗ A = µy∗ . Explain why y∗ x = 0 whenever λ = µ. 7.2.22. Consider A ∈ n×n . (a) Show that if A is diagonalizable, then there are right-hand and left-hand eigenvectors x and y∗ associated with λ ∈ σ (A) such that y∗ x = 0 so that we can make y∗ x = 1. (b) Show that not every right-hand and left-hand eigenvector x and y∗ associated with λ ∈ σ (A) must satisfy y∗ x = 0. (c) Show that (a) need not be true when A is not diagonalizable. Copyright c 2000 SIAM Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 524 Chapter 7 Eigenvalues and Eigenvectors http://www.amazon.com/exec/obidos/ASIN/0898714540 It is illegal to print, duplicate, or distribute this material Please report violations to [email protected] 7.2.23. Consider A ∈ n×n with λ ∈ σ (A) . (a) Prove that if λ is simple, then y∗ x = 0 for every pair of respective right-hand and left-hand eigenvectors x and y∗ associated with λ regardless of whether or not A is diagonalizable. Hint: Use the core-nilpotent decomposition on p. 397. (b) Show that y∗ x = 0 is possible when λ is not simple. 7.2.24. For A ∈ n×n with σ (A) = {λ1 , λ2 , . . . , λk } , show A is diagonalizable if and only if n = N (A − λ1 I) ⊕ N (A − λ2 I) ⊕· · ·⊕ N (A − λk I). Hint: Recall Exercise 5.9.14. D E 7.2.25. The Real Schur Form. Schur’s triangularization theorem (p. 508) insures that every square matrix A is unitarily similar to an uppertriangular matrix—say, U∗ AU = T. But even when A is real, U and T may have to be complex if A has some complex eigenvalues. However, the matrices (and the arithmetic) can be constrained to be real by settling for a block-triangular result with 2 × 2 or scalar entries on the diagonal. Prove that for each A ∈ n×...
View Full Document

This document was uploaded on 03/06/2014 for the course MA 5623 at City University of Hong Kong.

Ask a homework question - tutors are online