6
Exponential of a matrix
Definition 6.1. Given a square matrix A, we define its exponential eA as the series
∞
X Ak
A2 A3
e =I +A+
+
+ ... =
.
2!
3!
k!
k=0
A
It is easy to check that this series converges for any square matrix A whatsoever.
Theorem 6.2 (Constant matrices). If A is a constant square matrix, then Φ(t) = eAt is
a fundamental matrix for the system y 0 (t) = Ay. Moreover, the initial value problem
y 0 (t) = Ay,
y(0) = y0
has a unique solution which is defined for all times, namely y(t) = eAt y0 .
Example 6.3 (Computation of eAt ). When A is a diagonal matrix, we have




λ1
eλ1 t




At
..
..
A=
 =⇒ e = 
.
.
.
λn t
λn
e
When A is a 2 × 2 Jordan block, we have
¸
·
λ 1
=⇒
A=
λ
When A is a 3 × 3 Jordan block, we have


λ 1
A =  λ 1  =⇒
λ
·
At
e
¸
eλt teλt
=
.
eλt

eAt
eλt teλt
=
eλt

eλt
teλt  .
eλt
t2
2!
Using these facts, one can compute the exponential of any square matrix A. Namely, one
may determine the Jordan form P −1 AP and then use the formula
eAt = P · etP
−1 AP
· P −1
to relate the exponential of the given matrix A to that of its Jordan form.
Lemma 6.4 (Product rule). If A, B are square matrices, then (AB)0 = A0 B + AB 0 .
Lemma 6.5. If A, B are square matrices that commute, then eA+B = eA eB .
(6.1)