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

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: 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...
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