= Spectral decomposition of a quantum-plane point
{title2=$V=V_X\oplus V_Y\oplus U$}
For finite matrices satisfying $YX=qXY$, with $q$ not a <root of unity>, $Y$ sends the <generalized eigenspace> of $X$ at $\lambda$ into the one at $q^{-1}\lambda$. Finiteness of the <spectrum> makes $Y$ nilpotent on the sum of the nonzero $X$ <generalized eigenspaces>. Exchanging $X,Y$ gives the corresponding statement for $X$. Splitting the zero <generalized eigenspace> of $X$ by that of $Y$ gives the displayed invariant direct sum; both operators are nilpotent on $U$.
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