Let (in Description below) and (in Description below). Given that P diagonalizes A, then compute the matrix A^2517.

Question

Let

Given that P diagonalizes A, then compute the matrix A²⁵¹⁷.

Answer 1

When you are told that P diagonalizes A, it means that the product P^-1AP equals some diagonal matrix, D.

Equivalently, if P^-1AP = D, then:

AP = PD (premultiplying both sides of the above eqn by P)

A = PDP^-1 (postmultiplying by P^-1).

Now when you raise A to a power, the inner P and P^-1 terms will cancel, and the powers will accumulate on D:

A³

= AAA

= (PDP^-1)(PDP^-1)(PDP^-1)

= PDP^-1PDP^-1PDP^-1

= PD(P^-1P)D(P^-1P)DP^-1

= PDDDP^-1

= PD³P^-1.

So specific to this problem, A²⁵¹⁷ = P D²⁵¹⁷ P^-1. Raising diagonal matrices to a power then simply raises the diagonal elements to that power.

More generally, P is a matrix containing the eigenvectors of A, so the entries of D will be each vector's corresponding eigenvalues.