# Edu to prove 7912 establish that a i igi gi a i

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Unformatted text preview: λi I)ki . This coupled with the fact that dim N (A − λi I)ki ) = mi (Exercise 7.8.7) implies that Ji is a basis for R (Pi ) = N (A − λi I)ki . Consequently, each N (A − λi I)ki is an invariant subspace for A such that C n = N (A − λ1 I)k1 ⊕ N (A − λ2 I)k2 ⊕ · · · ⊕ N (A − λs I)ks It is illegal to print, duplicate, or distribute this material Please report violations to meyer@ncsu.edu and J(λi ) = A/ N D E . (A−λi I)ki Ji Of course, an even ﬁner direct sum decomposition of C n is possible because each Jordan segment is itself a block-diagonal matrix containing the individual Jordan blocks—the details are left to the interested reader. T H Exercises for section 7.8 7.8.1. Find the Jordan form of the following matrix whose distinct eigenvalues are σ (A) = {0, −1, 1}. Don’t be frightened by the size of A. ⎛ −4 −5 −3 1 −2 0 1 −2 ⎞ 4 7 3 2 6 −2 7 0 1 0 3 0 −1 2 5 0 −3 2 0 0 4 0 −1 3 −2 3 3 −4 −4 IG −1 ⎜ 0 −1 0 0 0 0 0 0⎟ ⎜ ⎟ ⎜ −1 1 2 −4 2 0 −3 1⎟ . A = ⎜ −8 −14 −5 1 −6 0 1 −4 ⎟ ⎜ ⎟ ⎝4 7 4 −3 3 −1 −3 4⎠ R Y P 7.8.2. For the matrix O C A= 1 −2 −1 that was used in Example 7.8.3, use the technique described on p. 594 to construct a nonsingular matrix P such that P−1 AP = J is in Jordan form. 7.8.3. Explain why index (λ) ≤ alg mult (λ) for each λ ∈ σ (An×n ) . 7.8.4. Explain why index (λ) = 1 if and only if λ is a semisimple eigenvalue. 7.8.5. Prove that every square matrix ⎛ similar to its transpose. Hint: Conis ⎞ 1 1 ⎜ sider the “reversal matrix” R = ⎝ . . . ⎟ ⎠ obtained by reversing the 1 order of the rows (or the columns) of the identity matrix I. Copyright c 2000 SIAM Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 7.8 Jordan Form http://www.amazon.com/exec/obidos/ASIN/0898714540 597 7.8.6. Cayley–Hamilton Revisited. Prove the the Cayley–Hamilton theorem (pp. 509, 532) by means of the J...
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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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