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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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