Matrix Exponential
2.10
2001/02/07
2002/10/24
Thanos
Antoulas
aca@rice.edu
John
Paul
Slavinsky
jps@alumni.rice.edu
Roy
Ha
rha@rice.edu
Elizabeth
Chan
lizychan@rice.edu
Thanos
Antoulas
aca@rice.edu
John
Paul
Slavinsky
jps@alumni.rice.edu
matrix exponential
system
Laplace transform
matrix
A formal description of the matrix exponential. The definition is given as
well as examples of calculating it. The matrix exponential in the
frequency domain is given as well through treatment by the Laplace
transform.
Since systems are often represented in terms of matrices and solutions
of system equations often make use of the exponential, it makes sense
to try and understand how these two concepts can be combined. In many
previous applications, we've seen terms like
a
t
come in handy for talking about system behavior. Here, a
was always a scalar quantity. However, what would happen if the scalar a was replaced by a matrix A?
The result would be what is known as the matrix exponential.
Definition
Recall the definition of the scalar exponential:
a
t
1
a
t
1
a
2
t
2
2
a
3
t
3
3
…
The definition of the matrix exponential is almost identical:
A
t
I
n
A
t
1
A
2
t
2
2
A
3
t
3
3
…
Where A is n x n and
I
n
is the n x n
identity matrix. While it is nice to see the resemblance between
these two definitions, applying this infinite series does not turn
out to be very efficient in practice. However, it can be useful
in certain special cases.
Compute
A
t
where
A
0
1
-1
0
.
We can start by taking powers of A
so that we can use the formal definition.
A
0
1
-1
0
A
2
0
1
-1
0
0
1
-1
0
-1
0
0
-1
I
A
3
A
2
A
A
A
4
A
2
A
2
I
A
5
A
A
2
A
A
6
A
2
A
4
I
And so the pattern goes, giving:
A
4
n
1
1
A
A
4
n
1
2
I
A
4
n
1
3
A
A
4
n
1
4
I
If we fill in the terms in the definition of
a
t
,
we'll get the following matrix:
A
t
1
t
2
2
t
4
4
…
t
t
3
3
t
5
5
…
t
t
3
3
t
5
5
…
1
t
2
2
t
4
4
…
We notice that the sums in this matrix look familiar-in fact,
they are the Taylor Series expansions of the sinusoids.
Therefore, the solution further reduces to:
A
t
t
t
t
t
General Method
The example above illustrates how the use of the true definition to
simplify matrix exponentials might only be easily applied in cases
with inherent repetition. There is a more general method involving
the Laplace Transform. In particular,
ℒ
A
t
s
I
A
We can verify that this is true by inserting the formal definition
of the matrix exponential:
ℒ
A
t
ℒ
I
A
t
1
A
2
t
2
2
…
1
s
I
1
s
2
A
1
s
3
A
2
…
s
I
A
The jump between the third and fourth equations here may be a
bit hard to believe, but this equality reduces to
I
I
when both sides are multiplied by
s
I
A
.
Taking an inverse Laplace of each side of Laplace Transform of the equation we find an expression for the matrix exponential:
A
t
ℒ
s
I
A
We can do the same example as before, this time using the
Laplace-based method.
A
0
1
-1
0
s
I
A
s
-1
1
s
1
s
2
1
s
1
-11
s
s
s
2
1
1
s
2
1
-1
s
2
1
s
s
2
1
Taking the inverse laplace of this gives us
A
t
t
t
t
t
Properties of the Matrix Exponential
In the scalar case, a product of exponentials
a
b
reduces to a single exponential whose power is the sum of the
individual exponents' powers,
a
b
.
However, in the case of the matrix exponential, this is not
true. If A and B
are matrices,
A
B
A
B
unless A and B
are commutative (i.e.
A
B
B
A
)
The derivative operates on the matrix exponential the same as
it does on the scalar exponential.
t
A
t
0
A
A
2
t
1
A
3
t
2
2
…
A
I
A
t
1
A
2
t
2
2
…
A
A
t