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

        [Graphics:Images/MatrixExponentialMod_gr_59.gif]  
and
        [Graphics:Images/MatrixExponentialMod_gr_60.gif]  
in general
        [Graphics:Images/MatrixExponentialMod_gr_61.gif]

Fact.  For a  [Graphics:Images/MatrixExponentialMod_gr_62.gif]  matrix this is

        [Graphics:Images/MatrixExponentialMod_gr_63.gif]
        
which can be simplified

        [Graphics:Images/MatrixExponentialMod_gr_64.gif]

The general case of  [Graphics:../Images/MatrixExponentialMod_gr_65.gif]

For the general case of an  [Graphics:../Images/MatrixExponentialMod_gr_66.gif]  matrix which has a full set of eigenpairs  [Graphics:../Images/MatrixExponentialMod_gr_67.gif]   for   [Graphics:../Images/MatrixExponentialMod_gr_68.gif],  the matrix power  [Graphics:../Images/MatrixExponentialMod_gr_69.gif] is given by

[Graphics:../Images/MatrixExponentialMod_gr_70.gif]


[Graphics:../Images/MatrixExponentialMod_gr_71.gif][Graphics:../Images/MatrixExponentialMod_gr_72.gif][Graphics:../Images/MatrixExponentialMod_gr_73.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2004