Fall 2003 Math 308/501–502
9 Matrix Methods for Linear Systems
9.8 The Matrix Exponential Function
Wed, 03/Dec
c
±
2003, Art Belmonte
Summary
In the following, let
A
and
B
be (real)
n
×
n
constant matrices, let
r
be an eigenvalue of
A
,andlet
v
be an eigenvector associated
with
r
(unless otherwise stated). Also, let
s
and
t
be scalars.
Deﬁnitions
•
The
exponential of the matrix A
is deﬁned by the power
series
e
A
=
I
+
A
+
1
2!
A
2
+
1
3!
A
3
+···=
∞
X
k
=
0
1
k
!
A
k
.
•
Similarly,
e
t
A
=
I
+
t
A
+
t
2
2!
A
2
+
t
3
3!
A
3
∞
X
k
=
0
t
k
k
!
A
k
.
•
If
(
A

r
I
)
p
w
=
0
, the zero vector, for some integer
p
≥
1,
then
w
is called a
generalized eigenvector
of
A
.
Properties of the matrix exponential
•
e
A
0
=
e
0
=
I
, the identity matrix.
•
e
(
s
+
t
)
A
=
e
s
A
e
t
A
.
•
(
e
t
A
)

1
=
e

t
A
.
•
If
A
and
B
commute (if
AB
=
BA
), then
e
t
(
A
+
B
)
=
e
t
A
e
t
B
.
•
e
st
I
=
e
I
.
Facts
1.
d
dt
e
t
A
=
A
e
t
A
. (Differentiate termbyterm to see this.)
2. Let
x
0
∈
R
n
.Then
x
(
t
)
=
e
t
A
x
0
is the unique solution to the
initial value problem
x
0
=
Ax
,
x
(
0
)
=
x
0
.
3. If
r
↔
v
is an eigenpair, then
e
t
A
v
=
e
rt
v
.
4.
A
commutes with
e
A
; i.e.,
A
e
A
=
e
A
A
.
5. If
r
is the
only
eigenvalue of
A
, then for some integer
m
<
n
we have
e
t
A
=
e
m
X
k
=
0
t
k
k
!
(
A

r
I
)
k
.
6. If
r
is an eigenvalue of
A
with algebraic multiplicity
q
,then
there is a positive integer
p
≤
q
such that the dimension of
the nullspace of
(
A

r
I
)
p
is equal to
q
.
7. Let
X
=
X
(
t
)
and
Y
=
Y
(
t
)
be fundamental matrices for
u
0
=
Au
and deﬁne
X
0
=
X
(
0
)
.
•
X
=
YC
for some constant matrix
C
.
•
e
t
A
=
X
(
t
)
X

1
0
: This gives another way to compute
e
t
A
.
•
The exponential matrix
e
t
A
itself
is a fundamental
matrix!
Full load of generalized eigenvectors
In light of #6, we ﬁnally have a procedure for
always
generating a
fundamental solution set to
x
0
=
Ax
. For each eigenvalue
r
of
A
,
let
q
be the algebraic multiplicity of
r
. Do the following.
•
Find the smallest positive integer
p
such that the nullspace of
(
A

r
I
)
p
has dimension
q
.
•
Find a basis
±
v
1
,...
v
q
²
for said nullspace.
•
For each generalized eigenvector
v
j
,
1
≤
j
≤
q
, construct
the solution
x
j
(
t
)
=
e
t
A
v
j
=
e
p

1
X
k
=
0
t
k
k
!
(
A

r
I
)
k
v
j
.
•
If
x
j
is associated with a complex eigenvalue
r
=
α
+
i
β
having positive imaginary part (i.e.,
β>
0), take its real and
imaginary parts to obtain two real solutions. (Toss out the
conjugate eigenvalue
α

i
β
.)
The collection of all these solutions over the selected eigenvalues
of
A
gives a real fundamental solution set.
MOAF (The Mother of All Formulas)
•
The solution of the IVP
x
0
=
Ax
+
f
,
x
(
t
0
)
=
x
0
, provided
A
is a constant matrix, is given by
x
(
t
)
=
e
(
t

t
0
)
A
x
0
+
Z
t
t
0
e
(
t

w)
A
f
(w)
d
w.
•
If
t
0
=
0, this simpliﬁes to
x
(
t
)
=
e
t
A
x
0
+
Z
t
0
e
(
t

A
f
(w)
d
1