**Unformatted text preview: **in the proof on p. 547 become real when “normal” is
replaced by “real-symmetric”). Consequently, A is real-symmetric if and only
if A is orthogonally similar to a real-diagonal matrix D. Below is a summary. D
E Symmetric and Hermitian Matrices In addition to the properties inherent to all normal matrices,
• Real-symmetric and hermitian matrices have real eigenvalues. (7.5.3) T
H • A is real symmetric if and only if A is orthogonally similar to a
real-diagonal matrix D —i.e., PT AP = D for some orthogonal P. • Real skew-symmetric and skew-hermitian matrices have pure imaginary eigenvalues. IG
R Example 7.5.1 Largest and Smallest Eigenvalues. Since the eigenvalues of a hermitian matrix An×n are real, they can be ordered as λ1 ≥ λ2 ≥ · · · ≥ λn . Y
P Problem: Explain why the largest and smallest eigenvalues can be described as
λ1 = max x∗ Ax
x λn = min x∗ Ax. and 2 =1 x (7.5.4) 2 =1 Solution: There is a unitary U such that U∗ AU = D = diag (λ1 , λ2 , . . . , λn )
or, equivalently, A = UDU∗ . Since x 2 = 1 ⇐⇒ y 2 = 1 for y = U∗ x, O
C x 2 =1 y 2 =1 n n max x∗ Ax = max y∗ Dy = max
y 2 =1 λi |yi |2 ≤ max λ1
y i=1 2 =1 |yi |2 = λ1
i=1 with equality being attained when x is an eigenvector of unit norm associated
with λ1 . The expression for the smallest eigenvalue λn is obtained by writing
n min x∗ Ax = min y∗ Dy = min x 2 =1 y 2 =1 y 2 =1 n λi |yi |2 ≥ min λn
y i=1 2 =1 |yi |2 = λn ,
i=1 where equality is attained at an eigenvector of unit norm associated with λn .
Note: The characterizations in (7.5.4) often appear in the equivalent forms
λ1 = max
x=0 Copyright c 2000 SIAM x∗ Ax
x∗ x and λn = min
x=0 x∗ Ax
.
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Chapter 7
Eigenvalues and Eigenvectors
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