= Diagonalizability criterion using the minimal polynomial
{title2=$m_T(t)=\prod_j(t-\lambda_j)\text{ with distinct }\lambda_j$}
= Minimal polynomial criterion for diagonalizability
{synonym}
A finite-dimensional <linear operator> over a <field> is diagonalizable exactly when its <minimal polynomial> splits into distinct linear factors. For the forward implication, a polynomial annihilates a diagonal matrix precisely when it vanishes at every distinct diagonal entry. Conversely, if $m_T=\prod_j(t-\lambda_j)$ with distinct roots, the <Chinese remainder theorem> yields a direct sum of the kernels of $T-\lambda_jI$. Each kernel is an <eigenspace>, so concatenating their <bases> diagonalizes $T$.
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