Lecture Notes in Differential Equations - Bruce E. Shapiro

310 LESSON 31. LINEAR SYSTEMS
Example 31.2. Solve the system
(
y ′ 6
= Ay where y(0) =
5)
and
From theorem 31.4
y = e A(t−t0) y 0 =
A =
( ) 1 1
4 −2
{ [( 1 1
exp
4 −2
) ]} ( 6
t
5
)
(31.65)
(31.66)
(31.67)
The problem now is that we don’t know how to calculate the matrix exponential!
We will continue this example after we have discussed some of its
properties.
Theorem 31.5. Properties of the Matrix Exponential.
1. If 0 is a square n × n matrix composed entirely of zeros, then
e 0 = I (31.68)
2. If A and B are square n×n matrices that commute (i.e., if AB = BA),
then
e A+B = e A e B (31.69)
3. e A is invertible, and
(
e
A ) −1
= e
−A
(31.70)
4. If A is any n × n matrix and S is any non-singular n × n matrix, then
S −1 e A S = exp ( S −1 AS ) (31.71)
5. Let D = diag(x 1 , ..., x n ) be a diagonal matrix. Then
e D = diag(e x1 , ..., e xn ). (31.72)
6. If S −1 AS = D, where D is a diagonal matrix, then e A = Se D S −1 .
7. If A has n linearly independent eigenvectors x 1 , ..., x n with corresponding
eigenvalues λ 1 , ..., λ n , then e A = UEU −1 , where U =
(x 1 , ..., x n ) and E = diag ( e λ1 , ..., e λn )
.
8. The matrix exponential is differentiable and d dt eAt = Ae At .