Example 1.  Consider the matrix  [Graphics:Images/MatrixExponentialMod_gr_80.gif],
1 (a)  Find  [Graphics:Images/MatrixExponentialMod_gr_81.gif].

Solution 1 (a).

We want to find

    [Graphics:../Images/MatrixExponentialMod_gr_83.gif]  

First look at some powers   [Graphics:../Images/MatrixExponentialMod_gr_84.gif]  

[Graphics:../Images/MatrixExponentialMod_gr_85.gif]



[Graphics:../Images/MatrixExponentialMod_gr_86.gif]

Now use the calculation   [Graphics:../Images/MatrixExponentialMod_gr_87.gif]  

[Graphics:../Images/MatrixExponentialMod_gr_88.gif]



[Graphics:../Images/MatrixExponentialMod_gr_89.gif]

Find the expression for the general term   [Graphics:../Images/MatrixExponentialMod_gr_90.gif]  

[Graphics:../Images/MatrixExponentialMod_gr_91.gif]



[Graphics:../Images/MatrixExponentialMod_gr_92.gif]

Find matrix exponential   [Graphics:../Images/MatrixExponentialMod_gr_93.gif]  will be the sum of the infinite series

[Graphics:../Images/MatrixExponentialMod_gr_94.gif]

The sum of the first few terms are:

[Graphics:../Images/MatrixExponentialMod_gr_95.gif]

[Graphics:../Images/MatrixExponentialMod_gr_96.gif]


[Graphics:../Images/MatrixExponentialMod_gr_97.gif]

[Graphics:../Images/MatrixExponentialMod_gr_98.gif]


[Graphics:../Images/MatrixExponentialMod_gr_99.gif]

[Graphics:../Images/MatrixExponentialMod_gr_100.gif]


[Graphics:../Images/MatrixExponentialMod_gr_101.gif]

[Graphics:../Images/MatrixExponentialMod_gr_102.gif]

Each element in   [Graphics:../Images/MatrixExponentialMod_gr_103.gif]  can be calculated by the sum of an infinite series and Mathematica can assist us in these computations.

[Graphics:../Images/MatrixExponentialMod_gr_104.gif]

[Graphics:../Images/MatrixExponentialMod_gr_105.gif]


[Graphics:../Images/MatrixExponentialMod_gr_106.gif]

[Graphics:../Images/MatrixExponentialMod_gr_107.gif]


[Graphics:../Images/MatrixExponentialMod_gr_108.gif]

[Graphics:../Images/MatrixExponentialMod_gr_109.gif]


[Graphics:../Images/MatrixExponentialMod_gr_110.gif]

[Graphics:../Images/MatrixExponentialMod_gr_111.gif]

Therefore, the matrix exponential   [Graphics:../Images/MatrixExponentialMod_gr_112.gif]  is

[Graphics:../Images/MatrixExponentialMod_gr_113.gif]


[Graphics:../Images/MatrixExponentialMod_gr_114.gif]

This can be compared to the matrix exponential  [Graphics:../Images/MatrixExponentialMod_gr_115.gif]  that can be computed by using Mathematica's built in procedure  MatrixExp[At].  

[Graphics:../Images/MatrixExponentialMod_gr_116.gif]



[Graphics:../Images/MatrixExponentialMod_gr_117.gif]

Caveat.  This shows the power of "artificial intelligence" that is available in Mathematica.  For this example the matrix had a full set of eigenvectors.  Mathematica is "smart" enough to know how to compute the matrix exponential for the difficult case when the set of eigenvectors is deficient.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2004