= Jordan–Chevalley decomposition
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{title2=$X=X_s+X_n,\quad[X_s,X_n]=0$}
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= Additive Jordan decomposition
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Over an <algebraically closed field>, a <linear operator> $X$ has a unique decomposition into a commuting <diagonalisable endomorphism> $X_s$ and <nilpotent endomorphism> $X_n$. On its <generalized eigenspace> for $\lambda$, set $X_s=\lambda I$ and $X_n=X-\lambda I$. The <Chinese remainder theorem> makes both parts polynomials in $X$, so they preserve every $X$-invariant subspace. Uniqueness follows by restricting any other commuting decomposition to those generalized eigenspaces. Over a perfect field the semisimple part need only become diagonalizable after scalar extension.
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