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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
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7.2 Diagonalization by Similarity Transformations
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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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