# Write the system of dierential equations in 741 on p

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Unformatted text preview: for a moment that f (z ) is a function from C into C that has a Taylor series expansion about λ. That is, for some r > 0, f (z ) = f (λ)+f (λ)(z −λ)+ f (λ) f (λ) (z −λ)2 + (z −λ)3 + · · · 2! 3! |z −λ| < r. for It is illegal to print, duplicate, or distribute this material Please report violations to [email protected] The representation (7.3.7) on p. 527 suggests that f (J ) should be deﬁned as f (J ) = f (λ)I + f (λ)(J − λI) + f (λ) f (λ) (J − λI)2 + (J − λI)3 + · · · . 2! 3! D E But since N = J − λI is nilpotent of index k, this series is just the ﬁnite sum k−1 f (J ) = i=0 T H f (i) (λ) i N, i! (7.9.1) and this means that only f (λ), f (λ), . . . , f (k−1) (λ) are required to exist. Also, ⎛0 ⎜ N=⎝ .. ⎛0 ⎞ 1 . .. .. . . ⎜ ⎟ 2⎜ ⎠, N = ⎜ ⎝ 1 0 .. IG ⎞ 1 .. . .. . . .. . .. . R Y 0 0 ⎛0 ⎟ ⎟ k−1 ⎜ =⎝ 1 ⎟, . . . , N ⎠ 0 0 .. ··· . .. . 1⎞ . .⎟ . 0 ⎠, 0 0 so the representation of f (J ) in (7.9.1) can be elegantly expressed as follows. P O C Functions of Jordan Blocks For a k × k Jordan block J with eigenvalue λ, and for a function f (z ) such that f (λ), f (λ), . . . , f (k−1) (λ) exist, f (J ) is deﬁned to be ⎛λ ⎜ f (J ) = f ⎝ ⎛ . f (λ) ⎞ 1 .. ⎜ ⎜ ⎜ ⎜ ⎟⎜ ⎠=⎜ ⎜ 1 ⎜ ⎜ λ ⎜ ⎝ f (λ) .. . .. . f (λ) f (λ) f (k−1) (λ) ··· 2! (k − 1)! f (λ) .. . .. .. . . . . . f (λ) 2! f (λ) f (λ) ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ . (7.9.2) ⎟ ⎟ ⎟ ⎟ ⎠ f (λ) Copyright c 2000 SIAM Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 7.9 Functions of Nondiagonalizable Matrices http://www.amazon.com/exec/obidos/ASIN/0898714540 Every Jordan form J = .. .J .. 601 is a block-diagonal matrix composed of . various Jordan blocks J , so (7.9.2) allows us to deﬁne f (J) = .. . (J ) f .. . as It is illegal to print, duplicate, or distribute this material Please report violations to [email protected] long as we pay attention to the fact that a suﬃcient number of derivatives of f are required to exist at the various eigenvalues. More precisely, if the s...
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