Powers of a diagonalisable matrix

Summary

Let $F$ be a field, and let $A$ be an $n \times n$ matrix over $F$ with $A = P D P^{-1}$. Here $P$ is invertible and $D$ is diagonal with entries $\lambda_1, \dots, \lambda_n$. Then, for each integer $k \ge 0$, $$A^k = P D^k P^{-1},$$ where $D^k$ is the diagonal matrix with entries $\lambda_1^k, \dots, \lambda_n^k$. So the powers of a diagonalisable matrix are easy to compute.