*This preview shows
page 1. Sign up
to
view the full content.*

**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