c ba c b nn o c are given by j b 2a ca cos j n1 and

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Unformatted text preview: 1. But this is impossible because the eigenvalues are distinct. Therefore, Mj = 0 for each j, and thus B is linearly independent. These results lead to the following characterization of diagonalizability. Diagonalizability and Multiplicities A matrix An×n is diagonalizable if and only if geo multA (λ) = alg multA (λ) (7.2.5) for each λ ∈ σ (A) —i.e., if and only if every eigenvalue is semisimple. Copyright c 2000 SIAM Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 7.2 Diagonalization by Similarity Transformations http://www.amazon.com/exec/obidos/ASIN/0898714540 513 It is illegal to print, duplicate, or distribute this material Please report violations to meyer@ncsu.edu Proof. Suppose geo multA (λi ) = alg multA (λi ) = ai for each eigenvalue λi . If there are k distinct eigenvalues, and if Bi is a basis for N (A − λi I), then k B = B1 ∪ B2 ∪ · · · ∪ Bk contains i=1 ai = n vectors. We just proved in (7.2.4) that B is a linearly independent set, so B represents a complete set of linearly independent eigenvectors of A, and we know this insures that A must be diagonalizable. Conversely, if A is diagonalizable, and if λ is an eigenvalue for A with alg multA (λ) = a, then there is a nonsingular matrix P such that P−1 AP = D = λIa×a 0 0 B , D E where λ ∈ σ (B). Consequently, / rank (A − λI) = rank P 0 0 0 B − λI P−1 = rank (B − λI) = n − a, T H and thus geo multA (λ) = dim N (A − λI) = n − rank (A − λI) = a = alg multA (λ) . IG R Example 7.2.4 Problem: Determine if either of the following matrices is diagonalizable: ⎛ ⎞ ⎛ ⎞ −1 −1 −2 1 −4 −4 A = ⎝ 8 −11 −8 ⎠ , B = ⎝ 8 −11 −8 ⎠ . −10 11 7 −8 8 5 Y P Solution: Each matrix has exactly the same characteristic equation λ3 + 5λ2 + 3λ − 9 = (λ − 1)(λ + 3)2 = 0, O C so σ (A) = {1, −3} = σ (B) , where λ = 1 has algebraic multiplicity 1 and λ = −3 has algebraic multiplicity 2. Since geo multA (−3) = dim N (A + 3I) = 1 < alg multA (−3) , A is not diagonalizable. On the other hand, geo multB (−3) = dim N (B + 3I) = 2 = alg multB (−3)...
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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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