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  "body": "\n## Statement\n\nLet $F$ be a field and let $A$ be an $n \\times n$ matrix over $F$.\nThen $A$ is [[similar|0E89HQATM8]] to an [[upper triangular matrix|0N4XY7Z3NN]] over $F$ if and only if the [[characteristic polynomial|0VSSPAFM2C]] of $A$ splits over $F$.\nHere, to split means to be a product of linear factors.\nIn particular, over an algebraically closed field, such as $\\C$, each square matrix is similar to an upper triangular matrix.\n\n## Remarks\n\nThe diagonal entries of the triangular matrix are the eigenvalues of $A$, with their algebraic multiplicities.\nThe proof is by induction: choose an eigenvector, extend it to a basis, and work on the quotient space.\nOver $\\C$, the matrix $P$ can even be chosen unitary; this is the Schur decomposition.\n\n## Explanations\n\n### L1 human\n\nNot every matrix can be diagonalised, but over $\\C$ every matrix can be made triangular by a change of basis.\nA triangular form is almost as useful: its eigenvalues are on the diagonal.\nOver $\\R$, a rotation matrix cannot be triangularised, because it has no real eigenvalue.\n",
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