Diagonalization expresses a square matrix as $A=PDP^{-1}$ with $P$ invertible and $D$ diagonal. Equivalently, $P^{-1}AP=D$. The columns of $P$ are an eigenvector basis, and the diagonal of $D$ contains their eigenvalues in the same order. It simplifies powers: $A^k=PD^kP^{-1}$.
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