## Monday, February 1, 2010

### Matrix Exponential

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.

.