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  

For the general case of an    matrix which has a full set of eigenpairs     for   ,  the matrix power   is given by

(c) John H. Mathews 2004