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Unformatted text preview: y (A − λI)−1 c = c/(λk − λ).
(b) For an arbitrary vector dn×1 , prove that the eigenvalues of
A + cdT agree with those of A except that λk is replaced by
λk + dT c.
(c) How can d be selected to guarantee that the eigenvalues of
A + cdT and A agree except that λk is replaced by a speciﬁed
number µ? Copyright c 2000 SIAM Buy online from SIAM
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7.1 Elementary Properties of Eigensystems
http://www.amazon.com/exec/obidos/ASIN/0898714540 503 7.1.18. Suppose that A is a square matrix.
(a) Explain why A and AT have the same eigenvalues.
(b) Explain why λ ∈ σ (A) ⇐⇒ λ ∈ σ (A∗ ) .
Hint: Recall Exercise 6.1.8.
(c) Do these results imply that λ ∈ σ (A) ⇐⇒ λ ∈ σ (A) when A
is a square matrix of real numbers?
(d) A nonzero row vector y∗ is called a left-hand eigenvector for
A whenever there is a scalar µ ∈ C such that y∗ (A − µI) = 0.
Explain why µ must be an eigenvalue for A in the “right-hand”
sense of the term when A is a square matrix of real numbers. D
E 7.1.19. Consider matrices Am×n and Bn×m .
(a) Explain why AB and BA have the same characteristic polynomial if m = n. Hint: Recall Exercise 6.2.16.
(b) Explain why the characteristic polynomials for AB and BA
can’t be the same when m = n, and then explain why σ (AB)
and σ (BA) agree, with the possible exception of a zero eigenvalue. T
R 7.1.20. If AB = BA, prove that A and B have a common eigenvector.
Hint: For λ ∈ σ (A) , let the columns of X be a basis for N (A − λI)
so that (A − λI)BX = 0. Explain why there exists a matrix P such
that BX = XP, and then consider any eigenpair for P. Y
P 7.1.21. For ﬁxed matrices Pm×m and Qn×n , let T be the linear operator on
C m×n deﬁned by T(A) = PAQ.
(a) Show that if x is a right-hand eigenvector for P and y∗ is a
left-hand eigenvector for Q, then xy∗ is an eigenvector for T.
(b) Explain why trace (T) = trace (P) trace (Q). O
C 7.1.22. Let D = diag (λ1 , λ2 , . . . ,...
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