12.1. EIGENVALUES AND EIGENVECTORS OF A MATRIX 265

then

Dm =

λ

m1 0

. . .

0 λmn

and that Am = SDmS−1as shown above. Recall why this was. A = SDS−1 and so

Am =

n times︷ ︸︸ ︷SDS−1SDS−1SDS−1 · · ·SDS−1 = SDmS−1

Now formally write the following power series for eA

eA ≡∞

∑k=0

Ak

k!=

∞

∑k=0

SDkS−1

k!= S

∞

∑k=0

Dk

k!S−1

If D is given above in 12.12, the above sum is of the form

S∞

∑k=0

1k! λ

k1 0

. . .

0 1k! λ

kn

S−1 = S

∑

∞k=0

1k! λ

k1 0

. . .

0 ∑∞k=0

1k! λ

kn

S−1

= S

eλ 1 0

. . .

0 eλ n

S−1

and this last thing is the definition of what is meant by eA.

Example 12.1.27 Let

A =

 2 −1 −11 2 1−1 1 2

Find eA.

The eigenvalues happen to be 1,2,3 and eigenvectors associated with these eigenvaluesare  −1

−11

↔ 2,

 0−11

↔ 1,

 −101

↔ 3

Then let

S =

 −1 0 −1−1 −1 01 1 1

and so

S−1 =

 −1 −1 −11 0 10 1 1

