But in addition to being real the eigenvalues of

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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 . x∗ x Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 550 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] Consequently, λ...
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