**Unformatted text preview: **d the diagonal entries of D are the associated eigenvalues. IG
R Proof. If A is normal with σ (A) = {λ1 , λ2 , . . . , λk } , then A − λk I is also normal. All normal matrices are RPN (range is perpendicular to nullspace, p. 409),
so there is a unitary matrix Uk such that
U∗ (A
k Y
P − λk I)Uk = O
C Ck
0 0
0 (by (5.11.15) on p. 408) or, equivalently U∗ AUk =
k Ck +λk I
0 0
λk I = Ak−1
0 0
λk I , where Ck is nonsingular and Ak−1 = Ck + λk I. Note that λk ∈ σ (Ak−1 ) (oth/
erwise Ak−1 − λk I = Ck would be singular), so σ (Ak−1 ) = {λ1 , λ2 , . . . , λk−1 }
(Exercise 7.1.4). Because Ak−1 is also normal, the same argument can be repeated with Ak−1 and λk−1 in place A and λk to insure the existence of a
unitary matrix Uk−1 such that
U∗ −1 Ak−1 Uk−1 =
k Copyright c 2000 SIAM Ak−2
0 0
λk−1 I , Buy online from SIAM
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548
Chapter 7
Eigenvalues and Eigenvectors
http://www.amazon.com/exec/obidos/ASIN/0898714540 It is illegal to print, duplicate, or distribute this material
Please report violations to [email protected] where Ak−2 is normal and σ (Ak−2 ) = {λ1 , λ2 , . . . , λk−2 } . After k such repetitions, Uk Uk−1 0 · · · U1 0 = U is a unitary matrix such that
0
I
0
I
⎛λ I
0
···
0⎞
1 a1
λ2 Ia2 · · ·
0⎟
⎜0
U∗ AU = ⎜ .
.
. ⎟ = D, ai = alg multA (λi ) . (7.5.1)
..
⎝.
.
.⎠
.
.
.
.
0
0
· · · λk Iak
Conversely, if there is a unitary matrix U such that U∗ AU = D is diagonal,
then A∗ A = UD∗ DU∗ = U = UDD∗ U∗ = AA∗ , so A is normal. D
E Caution! While it’s true that normal matrices possess a complete orthonormal
set of eigenvectors, not all complete independent sets of eigenvectors of a normal
A are orthonormal (or even orthogonal)—see Exercise 7.5.6. Below are some
things that are true. T
H Properties of Normal Matrices If A is a normal matrix with σ (A) = {λ1 , λ2 , . . . , λk } , then
• A is RPN—i.e., R (A) ⊥ N (A) (see p. 408).
•...

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