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Theorem C.0.1 Let f be given by 3.1 and suppose ρ (A) < R where R is the radius ofconvergence of the power series in 3.1. Then the series,

∞∑k=0

anAn (3.9)

converges in the space L (Fn,Fn) with respect to any of the norms on this space and further-more,

∞∑k=0

anAn = S−1

∑∞

n=0 anJn1 0

. . .

. . .

0∑∞

n=0 anJnr

S

where∑∞

n=0 anJnk is an mk ×mk matrix of the form given in 3.8 where A = S−1JS and

the Jordan form of A, J is given by 3.7. Therefore, you can define f (A) by the series in3.9.

Here is a simple example.

Example C.0.2 Find sin (A) where A =

4 1 −1 1

1 1 0 −1

0 −1 1 −1

−1 2 1 4

 .

In this case, the Jordan canonical form of the matrix is not too hard to find.4 1 −1 1

1 1 0 −1

0 −1 1 −1

−1 2 1 4

 =

2 0 −2 −1

1 −4 −2 −1

0 0 −2 1

−1 4 4 2

 ·

4 0 0 0

0 2 1 0

0 0 2 1

0 0 0 2



12

12 0 1

218 − 3

8 0 − 18

0 14 − 1

414

0 12

12

12

 .

Then from the above theorem sin (J) is given by

sin

4 0 0 0

0 2 1 0

0 0 2 1

0 0 0 2

 =

sin 4 0 0 0

0 sin 2 cos 2 − sin 22

0 0 sin 2 cos 2

0 0 0 sin 2

 .

Therefore, sin (A) =2 0 −2 −1

1 −4 −2 −1

0 0 −2 1

−1 4 4 2



sin 4 0 0 0

0 sin 2 cos 2 − sin 22

0 0 sin 2 cos 2

0 0 0 sin 2



12

12 0 1

218 − 3

8 0 − 18

0 14 − 1

414

0 12

12

12

 =M