Module

for

The Matrix Exponential

Solution of a Linear System of O. D. E.'s

Background for the Fundamental Matrix

We seek a solution of a homogeneous first order linear system of differential equations.  For illustration purposes we consider the case:

First, write the system in vector and matrix form

.

Then, find the eigenvalues and eigenvectors of the matrix  ,  denote the eigenpairs of  A  by

and   .

Assumption.  Assume that there are two linearly independent eigenvectors , which correspond to the eigenvalues , respectively.  Then two linearly independent solution to    are

,    and
.

Definition (Fundamental Matrix Solution)  The fundamental matrix solution  ,  is formed by using the two column vectors  .

(1)        .

The general solution to    is the linear combination

(2)        .

It can be written in matrix form using the fundamental matrix solution    as follows

.

Notation.  When we introduce the notation

,
and

The fundamental matrix solution    can be written as

(3)        .
or
(4)        .

The initial condition

If we desire to have the initial condition   ,  then this produces the equation

.

The vector of constant can be solved as follows

.

The solution with the prescribed initial conditions is

.

Observe that     where   is the identity matrix.  This leads us to make the following important definition

Definition (Matrix Exponential)  If    is a fundamental matrix solution to  ,  then the matrix exponential is defined to be

.

Notation. This can be written as

(5)        ,
or
(6)        .

Fact.  For a system, the initial condition is

,

and the solution with the initial condition   is

,
or
.

Theorem (Matrix Diagonalization)  The eigen decomposition of a   square matrix A is

,

which exists when A has a full set of eigenpairs     for   ,  and d is the diagonal matrix

and

is the augmented matrix whose columns are the eigenvectors of A.

.

Matrix power

How do you compute the higher powers of a matrix ?  For example, given
then
,
and
,  etc.

Exploration for

The higher powers seem to be intractable!  But if we have an eigen decomposition, then we are permitted to write

and

in general

Fact.  For a    matrix this is

which can be simplified

The general case of

Theorem (Series Representation for the Matrix Exponential)  The solution to   is given by the series

,    which becomes

and has the simplified form

,
or
.

Computer Programs  Matrix Exponential  Matrix Exponential

Example 1.  Consider the matrix  ,
1 (a)  Find  .
1 (b)  Find  .
Solution 1 (a).

Solution 1 (b).

Matrix Exponential 2D examples

The following examples illustrate the situation when there is a full set of eigenvectors.

Example 2.  Use the matrix exponential to find the general solution for the system of D. E.'s

Solution 2.

Example 3.  Use the matrix exponential to find the general solution for the system of D. E.'s       Solution 3.

Matrix Exponential 3D examples

The following examples illustrate the situation when there is a full set of eigenvectors.

Example 4.  Use the matrix exponential to find the general solution for the system of  D.E.'s   ,  where
.
Solution 4.

Example 5.  Use the matrix exponential to find the general solution for the system of  D.E.'s   ,  where
.
Solution 5.

Matrix Exponential 4D example

The following example illustrate the situation when there is a full set of eigenvectors.

Example 6.  Use the matrix exponential to find the general solution for the system of  D.E.'s  ,  where

Find the solution  ,that has the I.C.'s
.
Solution 6.

Matrix Exponential  Matrix Exponential  Internet hyperlinks to web sites and a bibliography of articles.

(c) John H. Mathews 2004