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Unformatted text preview: be f (A) = Pf (D)P−1 ⎛ f ( λ )I 0 ··· 1 f (λ2 )I · · · ⎜0 = P⎝ . . .. . . . 0 . 0 . ··· −1 ⎞ 0 0⎟ −1 . ⎠P . . . f ( λs ) I D E T H The Jordan decomposition A = PJP described on p. 590 easily provides a generalization of this idea to nondiagonalizable matrices. If J is the Jordan form for A, it’s natural to deﬁne f (A) by writing f (A) = Pf (J)P−1 . However, there are a couple of wrinkles that need to be ironed out before this notion actually makes sense. First, we have to specify what we mean by f (J)—this is not as clear as f (D) is for diagonal matrices. And after this is taken care of we need to make sure that Pf (J)P−1 is a uniquely deﬁned matrix. This also is not clear because, as mentioned on p. 590, the transforming matrix P is not unique—it would not be good if for a given A you used one P, and I used another, and this resulted in your f (A) being diﬀerent than mine. Let’s ﬁrst make sense of f (J). Assume throughout that A = PJP−1 ∈ C n×n with σ (A) = {λ1 , λ2 , . . . , λs } and where J = diag (J(λ1 ), J(λ2 ), . . . , J(λs )) is the Jordan form for A in which each segment J(λj ) is a block-diagonal matrix containing one or more Jordan blocks. That is, IG R Y P ⎛λ ⎞ 0 0 ⎟ . ⎠ . . . · · · Jtj(λj ) ⎛ J1(λ ) 0 · · · j ⎜ 0 J2(λj ) · · · J(λj ) = ⎝ . . .. . . O C . 0 . 0 j with ⎜ J (λj ) = ⎜ ⎝ 1 .. ⎞ . .. . .. . 1 λj ⎟ ⎟. ⎠ We want to deﬁne f (J) to be ⎛ ⎜ f (J) = ⎝ ⎛ ⎞ f J(λ1 ) .. ⎟ ⎠ . with ⎜ f J(λj ) = ⎝ .. ⎞ . .. f J(λs ) ⎟ ⎠, f J (λj ) . but doing so requires that we give meaning to f J (λj ) . To keep the notation λ from getting out of hand, let J = Copyright c 2000 SIAM 1 .. .. .. λ denote a generic k × k Jordan Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 600 Chapter 7 Eigenvalues and Eigenvectors http://www.amazon.com/exec/obidos/ASIN/0898714540 block, and let’s develop a deﬁnition of f (J ). Suppose...
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