= Diagonalizable restriction lemma
{title2=$T\text{ diagonalizable},\ T(W)\subseteq W\Longrightarrow T|_W\text{ diagonalizable}$}
If $W$ is an <invariant subspace> for a finite-dimensional <diagonalizable> <linear map> $T$, its restriction is <diagonalizable> over the same <field>. The <minimal polynomial> of $T|_W$ divides that of $T$: an annihilating <polynomial> for $T$ still annihilates the restriction, and <polynomial division> by its <minimal polynomial> proves divisibility. A divisor of a product of distinct <linear polynomials> remains a product of distinct <linear polynomials>, proving the assertion. This is the key step in <simultaneous diagonalization> of commuting <linear maps>.
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