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  "body": "\n## Statement\n\nLet $F$ be a field and let $A$ and $B$ be [[diagonalisable|07SQCNV233]] $n \\times n$ matrices over $F$.\nThen there is one invertible $n \\times n$ matrix $P$ such that both $P^{-1} A P$ and $P^{-1} B P$ are diagonal if and only if $AB = BA$.\n\n## Remarks\n\nDiagonal matrices commute, so the condition $AB = BA$ is necessary.\nFor sufficiency, $B$ maps each eigenspace of $A$ to itself, and it is diagonalisable on each of them.\nThe same holds for each family of pairwise commuting diagonalisable matrices.\n\n## Explanations\n\n### L1 human\n\nTwo diagonalisable matrices can be diagonalised by the same change of basis exactly when they commute.\nThen they share a basis of eigenvectors.\nThis fact is important in quantum mechanics, where commuting observables can be measured together.\n",
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