[GAP Forum] diagonalization of a matrix

[Walter Becker]

I am not sure if this is an appropriate question for the gap-forum 
but it might be an interesting math question anyway.
 
 The following four by four matrix reperesents the action of an order
five element on the elementary abelian group of order 11^4.   Call the 
following matrix e:
 
            [ 7 10  4  1]
            [ 5  1  9  7]
            [ 8  9  4  9]
            [ 9  8  8  5]
 
One reads this action as follows: 
      a^e*a^(-7)*b^(-10}*c^(-4)*d^(-1) =
      b^e*a^(-5)*b^(-1)*c^(-9)*d^(-7)=  
 
etc for c^e and d^e.  
 
The group generated by (a,b,c,,d,e) and 
with this action is equivalent (ie isomorphic)
to one in which the action by an
order 5 matrix takes the form:
 
a^e*a^(-x)=b^e*b^(-x)=c^e*c^(-y)=d^e*d^)-z)=1
 
where x=3, y=3^2, and z=3^3 mod(11).
 
This correspondence was found by asking for groups with the 
presentation in the form 
 
a^e*a^(-x)-b^e*b^(-y)=c^e*c^(-z)=d^e*d^(-w)=1.
 
Four cases of this type were found for the above given matrix. 
One knows that these exponents must be chosen from the set
of numbers 1,3,4,5,9.  
 
If a matrix is a real symmetric  (or hermitian) matrix then one knows
how to diagonalize it ---find its eigenvalues and eigenvectors and 
then the matrix of eigenvectors will enable one to get the diagonal form.
 
Does there exist a corresponding construction for a general matrix over
a finite field ?   Note here that even though the initial matrix is not 
symmetric it is equivalent to a diagonal matrix as far as the group is concerned.