11.5. FUNCTIONS OF MATRICES 301

for all |λ | < R. There is a formula for f (A) ≡ ∑∞n=0 anAn which makes sense whenever

ρ (A)< R. Thus you can speak of sin(A) or eA for A an n×n matrix. To begin with, define

fP (λ )≡P

∑n=0

anλn

so for k < P

f (k)P (λ ) =P

∑n=k

ann · · ·(n− k+1)λn−k =

P

∑n=k

an

(nk

)k!λ n−k. (11.17)

Thusf (k)P (λ )

k!=

P

∑n=k

an

(nk

n−k (11.18)

To begin with consider f (Jm (λ )) where Jm (λ ) is an m×m Jordan block. Thus Jm (λ ) =D+N where Nm = 0 and N commutes with D. Therefore, letting P > m

P

∑n=0

anJm (λ )n =P

∑n=0

an

n

∑k=0

(nk

)Dn−kNk =

P

∑k=0

P

∑n=k

an

(nk

)Dn−kNk

=m−1

∑k=0

NkP

∑n=k

an

(nk

)Dn−k. (11.19)

From 11.18 this equals

m−1

∑k=0

Nk diag

(f (k)P (λ )

k!, · · · ,

f (k)P (λ )

k!

)(11.20)

where for k = 0, · · · ,m−1, define diagk (a1, · · · ,am−k) the m×m matrix which equals zeroeverywhere except on the kth super diagonal where this diagonal is filled with the numbers,{a1, · · · ,am−k} from the upper left to the lower right. With no subscript, it is just thediagonal matrices having the indicated entries. Thus in 4× 4 matrices, diag2 (1,2) wouldbe the matrix 

0 0 1 00 0 0 20 0 0 00 0 0 0

 .

Then from 11.20 and 11.17,P

∑n=0

anJm (λ )n =m−1

∑k=0

diag k

(f (k)P (λ )

k!, · · · ,

f (k)P (λ )

k!

).

Therefore, ∑Pn=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 )

(11.21)

11.5. FUNCTIONS OF MATRICES 301for all |A| < R. There is a formula for f(A) = Y"_9a,A” which makes sense wheneverp (A) < R. Thus you can speak of sin (A) or e4 for A ann x n matrix. To begin with, definePA) = Yaa"n=0so fork <Pn—k P n ! n—k= Yn (n—k+l1)a = han A kan. (11.17)Thus ®)fe (A) GG (M\ a nki = han k A (11.18)To begin with consider f (Jj, (A)) where J, (A) is an m x m Jordan block. Thus Ji, (A) =D+WN where N” = 0 and N commutes with D. Therefore, letting P > mLowney = Yad ({)orint= ¥ Yan({) pewk=0n=kDxtye (i jor. k (11.19)From 11.18 this equalsm—1 (k) A (k) yy5 vais p_( ) de ( 7 (11.20)k! k!k=0where for k = 0,--- ,m—1, define diag, (a1,-+- ,@m—) the m x m matrix which equals zeroeverywhere except on the k’” super diagonal where this diagonal is filled with the numbers,{a1,+++ ,@m_—x} from the upper left to the lower right. With no subscript, it is just thediagonal matrices having the indicated entries. Thus in 4 x 4 matrices, diag, (1,2) wouldbe the matrixooo foooo eooo oFCo OoON OCThen from 11.20 and 11.17,P m—1 (k) a (k) 2Fda 2)" "Fe as (4 ).. et ).n=0 k=0 . .Therefore, YP) dnJm (A)" =TA (2) X (m—1) aSe (A) oe ny Joe asfo(a) “ee(2)f(a) rat (11.21)