hmmm its been a while since I worked with matrices (MATLAB to the rescue), but at first sight it seems like an eigenvalue problem. Im not sure if its possible to diaganolize the matrix, Im sorry its been a while:S.
take M = PDP^-1.
It should be possible to do this since you are dealing with a square matrix.
If you have done the diagonalization correctly you simple raise your lambda values to what ever k you wish to get M^k. This approach should be a start, if you havent covered matrix diagonalization by finding the eigenvectors of a matrix then Im not sure what other approach to take....but I dont think you need to do any diagonalization but simply consider the general theory of what is produced from it and consequently equate PDP^-1 with your summation.