A First Course in Linear Algebra, 2017a

386 Spectral Theory
7.3.5 The Matrix Exponential
The goal of this section is to use the concept of the matrix exponential to solve first order linear differential
equations. We begin by proving the matrix exponential.
Suppose A is a diagonalizable matrix. Then the matrix exponential,writtene A , can be easily defined.
Recall that if D is a diagonal matrix, then
P −1 AP = D
D is of the form
and it follows that
⎡
⎢
⎣
D m =
⎤
λ 1 0
. ..
⎥
⎦ (7.5)
0 λ n
⎡
⎢
⎣
λ m 1
0
. ..
0 λ m n
⎤
⎥
⎦
Since A is diagonalizable,
and
Recall why this is true.
and so
A = PDP −1
A m = PD m P −1
A = PDP −1
{
m times
}} {
A m = PDP −1 PDP −1 PDP −1 ···PDP −1
= PD m P −1
We now will examine what is meant by the matrix exponental e A . Begin by formally writing the
following power series for e A :
)
e A D k
=
P −1
k!
(
∞
A
∑
k ∞
k=0
k! = PD
∑ k P ∞∑ −1
= P
k=0
k!
k=0
If D is given above in 7.5, the above sum is of the form
⎛ ⎡
1
∞ k!
⎜ ⎢
λ 1 k 0
P⎝∑
⎣
. ..
k=0
0
This can be rearranged as follows:
⎡
e A ⎢
= P⎣
1
k! λ k n
∑ ∞ k=0 1 k! λ k 1
0
. ..
⎤⎞
⎥⎟
⎦
⎠P −1
0 ∑ ∞ k=0 1 k! λ k n
⎤
⎥
⎦P −1