# Now determine r g the image of any z is gz x with y

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Unformatted text preview: on p. 325. If P = x | X , then straightforward multiplication shows that O C P−1 = y∗ X (I − xy∗ ) ∗ and P−1 AP = λ 0 0 B , where B = X∗ AX is n − 1 × n − 1. The eigenvalues of B constitute the remaining eigenvalues of A (Exercise 7.1.4), and thus an n × n eigenvalue problem is deﬂated to become one of size n − 1 × n − 1. Note: When A is symmetric, we can take x = y to be an eigenvector with 0 x 2 = 1, so P = R = R−1 , and RAR = λ B in which B = BT . 0 An elegant and more geometrical way of expressing diagonalizability is now presented to help simplify subsequent analyses and pave the way for extensions. Copyright c 2000 SIAM Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 7.2 Diagonalization by Similarity Transformations http://www.amazon.com/exec/obidos/ASIN/0898714540 517 Spectral Theorem for Diagonalizable Matrices A matrix An×n with spectrum σ (A) = {λ1 , λ2 , . . . , λk } is diagonalizable if and only if there exist matrices {G1 , G2 , . . . , Gk } such that It is illegal to print, duplicate, or distribute this material Please report violations to [email protected] A = λ 1 G 1 + λ 2 G2 + · · · + λ k Gk , (7.2.7) where the Gi ’s have the following properties. • Gi is the projector onto N (A − λi I) along R (A − λi I). • Gi Gj = 0 whenever i = j. D E (7.2.8) (7.2.9) (7.2.10) • G1 + G2 + · · · + G k = I. The expansion (7.2.7) is known as the spectral decomposition of A, and the Gi ’s are called the spectral projectors associated with A. T H Proof. If A is diagonalizable, and if Xi is a matrix whose columns form a basis for N (A − λi I), then P = X1 | X2 | · · · | Xk is nonsingular. If P−1 is partitioned in a conformable manner, then we must have ⎛ T⎞ Y1 ⎛ ⎞⎜ ⎟ λ1 I 0 · · · 0 ⎜ T⎟ 0 λ2 I · · · 0 ⎟ ⎜ Y2 ⎟ ⎜ ⎟ ⎜ A = PDP−1 = X1 | X2 | · · · | Xk ⎜ . . . ⎟⎜ .. ⎝. . . ⎠⎜ . ⎟ ⎟ . . . . ⎜ . ⎟ (7.2.11) ⎝.⎠ 0 0 · · · λk I T Yk IG R Y P T T T = λ1 X 1 Y 1 + λ2 X 2 Y 2 + · · · + λ k Xk Y k O C = λ 1 G1 + λ 2 G 2 + · · · + λ k G k . T For Gi = Xi Yi , the statement P...
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