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  "body": "\n## Summary\n\nLet $F$ be a field, and let $A$ be an $n \\times n$ matrix over $F$ with $A = P D P^{-1}$.\nHere $P$ is invertible and $D$ is diagonal with entries $\\lambda_1, \\dots, \\lambda_n$.\nThen, for each integer $k \\ge 0$,\n$$A^k = P D^k P^{-1},$$\nwhere $D^k$ is the diagonal matrix with entries $\\lambda_1^k, \\dots, \\lambda_n^k$.\nSo the powers of a [[diagonalisable|07SQCNV233]] matrix are easy to compute.\n\n## Remarks\n\nThe factors $P^{-1} P$ in $(P D P^{-1})^k$ cancel in pairs.\nThe same idea gives $p(A) = P\\, p(D)\\, P^{-1}$ for each polynomial $p$.\nOver $\\R$ or $\\C$, it also gives the matrix exponential $e^A = P e^D P^{-1}$.\nIt is the standard method to solve linear recurrences, such as the Fibonacci recurrence.\n\n## Explanations\n\n### L1 human\n\nTo raise a diagonalisable matrix to a power, change to the basis of eigenvectors.\nThere the matrix is diagonal, and its power is the diagonal of the powers of the eigenvalues.\nThen change back to the old basis.\n",
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