I must be missing something as this seems very basic. Assume A = vEv^T , where E is a diagonal matrix of eigenvalues and v is the transformation from A to E. T

I must be missing something as this seems very basic. Assume A = vEv^T , where E is a diagonal matrix of eigenvalues and v is the transformation from A to E. Then A^n = vEv^TvEv^T..vEv^T. vv^T equals the identity matrix and will cancel out. So A^n = vE^nv^T. The closed form in the article is then just inlining the computation of v and v^T to how they apply to alpha and beta. Oh, and to understand what happens in the more complex case where not all eigenvalues are different, see Jordan Normal Form. That is less basic and not as well covered in the basic courses as the above is.