A First Course in Linear Algebra, 2017a

388 Spectral Theory
The matrix exponential is a useful tool to solve autonomous systems of first order linear differential
equations. These are equations which are of the form
X ′ = AX,X(0)=C
where A is a diagonalizable n × n matrix and C is a constant vector. X is a vector of functions in one
variable, t:
⎡ ⎤
x 1 (t)
x 2 (t)
X = X(t)= ⎢ ⎥
⎣ . ⎦
x n (t)
Then X ′ refers to the first derivative of X and is given by
⎡
x ′ 1 (t)
⎤
X ′ = X ′ x ′ 2
(t)= ⎢
(t)
⎥
⎣ . ⎦ , x′ i(t)=the derivative of x i (t)
x ′ n (t)
Then it turns out that the solution to the above system of equations is X (t)=e At C. To see this, suppose
A is diagonalizable so that
⎡
⎤
λ 1
⎢
A = P⎣
. ..
⎥
⎦P −1
λ n
♠
Then
⎡
e At ⎢
= P⎣
e λ 1t
. ..
e λ nt
⎤
⎥
⎦P −1
⎡
e At ⎢
C = P⎣
e λ 1t
. ..
e λ nt
⎤
⎥
⎦P −1 C
Differentiating e At C yields
X ′ =
⎡
( ) ′
e At ⎢
C = P ⎣
λ 1 e λ 1t
. ..
λ n e λ nt
⎤
⎥
⎦P −1 C
⎡
⎢
= P⎣
⎤⎡
λ 1
. ..
λ n
⎥⎢
⎦⎣
e λ 1t
. ..
e λ nt
⎤
⎥
⎦P −1 C