The Matrix Exponential
2.8
2002/06/21
2002/07/10 00:00:00.008 GMT-5
Steven
Cox
cox@rice.edu
Liqun
Wang
liqun@rice.edu
Steven
Cox
cox@rice.edu
matrix
exponential
This module describes the general idea of matrix exponential.
The matrix exponential is a powerful means for representing the
solution to n linear, constant
coefficient, differential equations. The initial value problem
for such a system may be written
x
t
A
x
t
x
0
x0
where A is the n-by-n matrix of
coefficients. By analogy to the 1-by-1 case we might expect
x
t
A
t
u
to hold. Our expectations are granted if we properly define
A
t
. Do you see why simply exponentiating each element of
At will no suffice?
There are at least 4 distinct (but of course equivalent)
approaches to properly defining
A
t
. The first two are natural analogs of the single
variable case while the latter two make use of heavier matrix
algebra machinery.
- The Matrix
Exponential as a Limit of Powers
- The Matrix
Exponential as a Sum of Powers
- The Matrix
Exponential via the Laplace Transform
- The Matrix
Exponential via Eigenvalues and Eigenvectors
Please visit each of these modules to see the definition and a
number of examples.
For a concrete application of these methods to a real dynamical
system, please visit the
Mass-Spring-Damper module.
Regardless of the approach, the matrix
exponential may be shown to obey the 3 lovely properties
-
t
A
t
A
A
t
A
t
A
-
A
t1
t2
A
t1
A
t2
-
A
t
is nonsingular and
A
t
A
t
Let us confirm each of these on the suite of examples used in
the submodules.
If
A
1
0
0
2
then
A
t
t
0
0
2
t
-
t
A
t
t
0
0
2
2
t
1
0
0
2
t
0
0
2
t
-
t1
t2
0
0
2
t1
2
t2
t1
t2
0
0
2
t1
2
t2
t1
0
0
2
t1
t2
0
0
2
t2
-
A
t
t
0
0
2
t
A
t
If
A
0
1
-1
0
then
A
t
t
t
t
t
-
t
A
t
t
t
t
t
and
A
A
t
t
t
t
t
-
You will recognize this statement as a basic trig
identity
t1
t2
t1
t2
t1
t2
t1
t2
t1
t1
t1
t1
t2
t2
t2
t2
-
Using the cofactor expansion we find
A
t
t
t
t
t
t
t
t
t
A
t
If
A
0
1
0
0
then
A
t
1
t
0
1
-
t
A
t
0
1
0
0
A
A
t
-
1
t1
t2
0
1
1
t1
0
1
1
t2
0
1
-
1
t
0
1
1
t
0
1
A
t