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n=0 cn (A − z0 I)
and expanding the result, the following result is established. Copyright c 2000 SIAM Buy online from SIAM
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7.3 Functions of Diagonalizable Matrices
http://www.amazon.com/exec/obidos/ASIN/0898714540 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
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
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
−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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