# B explain why n nt is nonsingular if and only if n is

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Unformatted text preview: tements are equivalent. • A is similar to a diagonal matrix—i.e., P−1 AP = D. • A has a complete linearly independent set of eigenvectors. • Every λi is semisimple—i.e., geo multA (λi ) = alg multA (λi ) . A = λ1 G1 + λ2 G2 + · · · + λk Gk , where Gi is the projector onto N (A − λi I) along R (A − λi I), • Y P O C Gi Gj = 0 whenever i = j, G 1 + G 2 + · · · + G k = I, k Gi = j =1 j =i (A − λj I) k (λi − λj ) (see (7.3.11) on p. 529). j =1 j =i If λi is a simple eigenvalue associated with right-hand and lefthand eigenvectors x and y∗ , respectively, then Gi = xy∗ /y∗ x. Exercises for section 7.2 7.2.1. Diagonalize A = −8 −6 with a similarity transformation, or else 12 10 explain why A can’t be diagonalized. 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 7.2.2. (a) Verify that alg multA (λ) = geo multA (λ) for each eigenvalue of ⎛ ⎞ −4 −3 −3 A = ⎝ 0 −1 0⎠. 6 6 5 (b) It is illegal to print, duplicate, or distribute this material Please report violations to [email protected] 521 Find a nonsingular P such that P−1 AP is a diagonal matrix. 7.2.3. Show that similar matrices need not have the same eigenvectors by giving an example of two matrices that are similar but have diﬀerent eigenspaces. 7.2.4. λ = 2 is an eigenvalue for A = 3 0 −2 2 2 −3 1 0 0 D E . Find alg multA (λ) as T H well as geo multA (λ) . Can you conclude anything about the diagonalizability of A from these results? 7.2.5. If B = P−1 AP, explain why Bk = P−1 Ak P. IG R 7/5 −1 7.2.6. Compute limn→∞ An for A = 1 /5 1 /2 . 7.2.7. Let {x1 , x2 , . . . , xt } be a set of linearly independent eigenvectors for An×n associated with respective eigenvalues {λ1 , λ2 , . . . , λt } , and let X be any n × (n − t) matrix such that Pn×n = x1 | · · · | xt | X is ⎛ ∗⎞ Y P y1 nonsingular. Prove that i...
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