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  "body": "\n## Statement\n\nLet $F$ be a perfect field, for example a field of characteristic $0$ or an algebraically closed field.\nLet $V$ be a finite-dimensional vector space over $F$, and let $T \\colon V \\to V$ be a linear map.\nThen there are unique linear maps $S, N \\colon V \\to V$ with these properties:\n\n1. $T = S + N$.\n2. $S$ is [[semisimple|08VRX3Q4PB]] and $N$ is [[nilpotent|063D5Z2C70]].\n3. $SN = NS$.\n\n## Remarks\n\nMoreover, $S$ and $N$ are polynomials in $T$ without constant term.\nOver $\\C$, $S$ is diagonalisable; in a Jordan basis, $S$ is the diagonal part of the Jordan form.\nFor $J_2(\\lambda)$, the decomposition is $S = \\lambda I_2$ and $N = \\begin{pmatrix} 0 \u0026 1 \\\\ 0 \u0026 0 \\end{pmatrix}$.\n\n## Explanations\n\n### L1 human\n\nEvery linear map over $\\C$ splits into a diagonalisable part and a nilpotent part.\nThe two parts commute with each other.\nThis splitting is unique, and it makes functions of matrices easier to compute.\n",
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