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.   

        .

Proof  Matrix Exponential  Matrix Exponential  

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
        .  

Proof  Matrix Exponential  Matrix Exponential  

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.

Research Experience for Undergraduates

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

Download this Mathematica Notebook The Matrix Exponential

(c) John H. Mathews 2004