302 CHAPTER 11. LIMITS OF VECTORS AND MATRICES

Now let A be an n×n matrix with ρ (A)< R where R is given above. Then the Jordan formof A is of the form

J =

J1 0

J2. . .

0 Jr

 (11.22)

where Jk = Jmk (λ k) is an mk×mk Jordan block and A = S−1JS. Then, letting P > mk forall k,

P

∑n=0

anAn = S−1P

∑n=0

anJnS,

and because of block multiplication of matrices,

P

∑n=0

anJn =

∑

Pn=0 anJn

1 0. . .

. . .

0 ∑Pn=0 anJn

r

and from 11.21 ∑

Pn=0 anJn

k converges as P→ ∞ to the mk×mk matrix

f (λ k)f ′(λ k)

1!f (2)(λ k)

2! · · · f (m−1)(λ k)(mk−1)!

0 f (λ k)f ′(λ k)

1!. . .

...

0 0 f (λ k). . . f (2)(λ k)

2!...

. . . . . . f ′(λ k)1!

0 0 · · · 0 f (λ k)

(11.23)

There is no convergence problem because |λ | < R for all λ ∈ σ (A) . This has proved thefollowing theorem.

Theorem 11.5.1 Let f be given by 11.16 and suppose ρ (A) < R where R is the radius ofconvergence of the power series in 11.16. Then the series,

∞

∑k=0

anAn (11.24)

converges in the space L (Fn,Fn) with respect to any of the norms on this space andfurthermore,

∞

∑k=0

anAn = S−1

∑

∞n=0 anJn

1 0. . .

. . .

0 ∑∞n=0 anJn

r

S