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R Eigenvectors corresponding to distinct eigenvalues are orthogonal. In
N (A − λi I) ⊥ N (A − λj I) • Y
P for λi = λj . (7.5.2) The spectral theorems (7.2.7) and (7.3.6) on pp. 517 and 526 hold,
but the spectral projectors Gi on p. 529 specialize to become orthogonal projectors because R (A − λi I) ⊥ N (A − λi I) for each λi . O
C Proof of (7.5.2). If A is normal, so is A − λj I, and hence A − λj I is RPN.
Consequently, N (A − λj I) = N (A − λj I) —recall (5.11.14) from p. 408. If
(λi , xi ) and (λj , xj ) are distinct eigenpairs, then (A − λj I)∗ xj = 0, and 0 =
x∗ (A − λj I)xi = x∗ Axi − x∗ λj xi = (λi − λj )x∗ xi implies 0 = x∗ xi .
Several common types of matrices are normal. For example, real-symmetric
and hermitian matrices are normal, real skew-symmetric and skew-hermitian
matrices are normal, and orthogonal and unitary matrices are normal. By virtue
of being normal, these kinds of matrices inherit all of the above properties, but
it’s worth looking a bit closer at the real-symmetric and hermitian matrices
because they have some special eigenvalue properties.
If A is real symmetric or hermitian, and if (λ, x) is an eigenpair for A,
then x∗ x = 0, and λx = Ax implies λx∗ = x∗ A∗ , so
x∗ x(λ − λ) = x∗ (λ − λ)x = x∗ Ax − x∗ A∗ x = 0 =⇒ λ = λ. Copyright c 2000 SIAM Buy online from SIAM
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7.5 Normal Matrices
http://www.amazon.com/exec/obidos/ASIN/0898714540 549 It is illegal to print, duplicate, or distribute this material
Please report violations to [email protected] In other words, eigenvalues of real-symmetric and hermitian matrices are real.
A similar argument (Exercise 7.5.4) shows that the eigenvalues of a real skewsymmetric or skew-hermitian matrix are pure imaginary numbers.
Eigenvectors for a hermitian A ∈ C n×n may have to contain complex numbers, but a real-symmetric matrix possesses a complete orthonormal set of real
eigenvectors (all quantities...
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