You notation is confusing. I know you mean but what do you mean by " "? I suspect you have combined two notations and really mean that "Ax"is.

If that is the case, then you need to find all eigenvalues and a corresponding eigenvector for each eigenvalue (the "eigenpairs").

Once you have that you can write where "P" is the matrix having eigevectors as columns and D is a diagonal matrix with the eigenvalues on the diagonal.

It is easy to find powers of a diagonal matrix: is just the diagonal matrix having the nth power of the eigenvalues on the diagonal. Further, , , etc.

where is just the diagonal matrix having on the diagonal with being the eigenvalues.