398 APPENDIX C. FUNCTIONS OF MATRICES

Then from 3.5 and 3.2,

P∑n=0

anJm (λ)n=

m−1∑k=0

diag k

(f(k)P (λ)

k!, · · · ,

f(k)P (λ)

k!

).

Therefore,∑P

n=0 anJm (λ)n=

fP (λ)f ′P (λ)1!

f(2)P (λ)

2! · · · f(m−1)P (λ)

(m−1)!

fP (λ)f ′P (λ)1!

. . ....

fP (λ). . . f

(2)P (λ)

2!

. . . f ′P (λ)1!

0 fP (λ)

(3.6)

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

J =

J1 0

J2. . .

0 Jr

 (3.7)

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

for all k,P∑

n=0

anAn = S−1

P∑n=0

anJnS,

and because of block multiplication of matrices,

P∑n=0

anJn =

∑P

n=0 anJn1 0

. . .

. . .

0∑P

n=0 anJnr

and from 3.6

∑Pn=0 anJ

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

f (λk)f ′(λk)

1!f(2)(λk)

2! · · · f(m−1)(λk)(mk−1)!

0 f (λk)f ′(λk)

1!

. . ....

0 0 f (λk). . . f(2)(λk)

2!...

. . .. . . f ′(λk)

1!

0 0 · · · 0 f (λk)

(3.8)

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

398 APPENDIX C. FUNCTIONS OF MATRICESThen from 3.5 and 3.2,n=0 k=0 RRTherefore, yo AnJm (A)" =, (2) (4 Gm OYfp (\) fe) fe do, ifp(r) 2 :; (2)fe(d) - f2M (3.6); fpQ)T!0 fp (A)Now let A be an n x n matrix with p(A) < R where R is given above. Then the Jordanform of A is of the formJi 0JoJ= ; (3.7)0 Jywhere Jy = Jm, (Ar) is an mg X mz Jordan block and A = S~!JS. Then, letting P > mz,for all k,P PSo an A" = S71 S° an JS,n=0 n=0and because of block multiplication of matrices,T= Ind} 0P .Ya" =n=0 .0 Tao On< P .and from 3.6 }7),) GnJj’ converges as P — oo to the mz x mz matrix, (2) (m-1)f Ar) f Ge) f (Ae) wee poe0 Ff (Ak) £Os) :0 0 fx) o fQw (3.8). f'n). . . 1!0 0 0 FOR)There is no convergence problem because |\| < R for all A € o (A). This has proved thefollowing theorem.