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At ﬁrst glance this deﬁnition seems to have an edge over the inﬁnite series approach because there are no convergence issues to deal with. But convergence
worries have been traded for uniqueness worries. Because P is not unique, it’s
not apparent that (7.3.4) is well deﬁned. The eigenvector matrix P you compute
for a given A need not be the same as the eigenvector matrix I compute, so what
insures that your f (A) will be the same as mine? The spectral theorem (p. 517)
does. Suppose there are k distinct eigenvalues that are grouped according to
repetition, and expand (7.3.4) just as (7.2.11) is expanded to produce
⎛ T⎞
Y1
⎛ f (λ )I
0
···
0 ⎞⎜ T ⎟
1
⎟
⎜
f (λ2 )I · · ·
0 ⎟⎜ Y2 ⎟
⎜0
⎟
⎜
−1
f (A) = PDP = X1 |X2 | · · · |Xk ⎜ .
.
. ⎟⎜
..
⎝.
. ⎠⎜ . ⎟
.
⎟
.
.
.
.
⎜.⎟
.⎠
0
0
· · · f (λk )I ⎝
k D
E T
H k
T
f (λi )Xi Yi = =
i=1 f (λi )Gi .
i=1 T
Yk IG
R Since Gi is the projector onto N (A − λi I) along R (A − λi I), Gi is uniquely
determined by A. Therefore, (7.3.4) uniquely deﬁnes f (A) regardless of the
choice of P. We can now make a formal deﬁnition. Functions of Diagonalizable Matrices Y
P Let A = PDP−1 be a diagonalizable matrix where the eigenvalues in
D = diag (λ1 I, λ2 I, . . . , λk I) are grouped by repetition. For a function
f (z ) that is deﬁned at each λi ∈ σ (A) , deﬁne O
C f (A) = Pf (D)P−1 ⎛ f (λ )I
1
⎜0
= P⎜ .
⎝.
.
0 0
f (λ2 )I
.
.
.
0 ···
···
..
. 0
0
.
.
. ⎞
⎟ −1
⎟ P (7.3.5)
⎠ · · · f (λk )I = f (λ1 )G1 + f (λ2 )G2 + · · · + f (λk )Gk , (7.3.6) where Gi is the ith spectral projector as described on pp. 517, 529.
The generalization to nondiagonalizable matrices is on p. 603.
The discussion of matrix functions was initiated by considering inﬁnite series, so, to complete the circle, a formal statement connecting inﬁnite series with
∞
n
(7.3.5) and (7.3.6) is needed. By replacing A by PDP−1...

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