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Unformatted text preview: in n=0 cn (A − z0 I) and expanding the result, the following result is established. Copyright c 2000 SIAM Buy online from SIAM Buy from 7.3 Functions of Diagonalizable Matrices 527 Infinite Series ∞ n=0 cn (z If f (z ) = − z0 )n converges when |z − z0 | < r, and if |λi − z0 | < r for each eigenvalue λi of a diagonalizable matrix A, then It is illegal to print, duplicate, or distribute this material Please report violations to [email protected] ∞ cn (A − z0 I)n . f (A) = (7.3.7) D E n=0 It can be argued that the matrix series on the right-hand side of (7.3.7) converges if and only if |λi −z0 | < r for each λi , regardless of whether or not A is diagonalizable. So (7.3.7) serves to define f (A) for functions with series expansions regardless of whether or not A is diagonalizable. More is said in Example 7.9.3 (p. 605). T H IG R Example 7.3.1 Neumann Series Revisited. The function f (z ) = (1 − z )−1 has the geometric ∞ series expansion (1 − z )−1 = k=0 z k that converges if and only if |z | < 1. This means that the associated matrix function f (A) = (I − A)−1 is given by (I − A)−1 = Y P ∞ Ak if and only if |λ| < 1 for all λ ∈ σ (A) . (7.3.8) k=0 This is the Neumann series discussed on p. 126, where it was argued that ∞ if limn→∞ An = 0, then (I − A)−1 = k=0 Ak . The two approaches are the n same because it turns out that limn→∞ A = 0 ⇐⇒ |λ| < 1 for all λ ∈ σ (A) . This is immediate for diagonalizable matrices, but the nondiagonalizable case is a bit more involved—the complete statement is developed on p. 618. Because maxi |λi | ≤ A for all matrix norms (Example 7.1.4, p. 497), a corollary of (7.3.8) is that (I − A)−1 exists and O C −1 (I − A) ∞ Ak = when A < 1 for any matrix norm. (7.3.9) k=0 Caution! (I − A)−1 can exist without the Neumann series expansion being valid because all that’s needed for I − A to be...
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