Matrix Exponential

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 ofAby

and.

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

,and
.

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

(1).

The general solution tois the linear combination

(2).

It can be written in matrix form using the fundamental matrix solutionas follows

.

Notation.When we introduce the notation

,
and


The fundamental matrix solutioncan 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 thatwhere is the identity matrix.This leads us to make the following important definition

Definition (Matrix Exponential)Ifis 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 eigenpairsfor,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.

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 amatrix this is



which can be simplified

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

,which becomes



and has the simplified form

,
or
.