Your language is a bit confusing. You titled this "Generalizing the n-th power of a matrix" but then talk about a vector- except that the example you show is a matrix! And why talk about "generalizing the n-th power"? What you are really asking about iscalculatingthe n-th power of a matrix, there is no "generalization" involved.

I don't know what you mean by using the "z-transform" to find powers but diagonalizing certainly will work. Since the matrix you give it has three distinct eigenvalues and can be diagonalized.