Diagonalizability criterion using the minimal polynomial ($m_T(t)={\prod }_j(t-{\lambda }_j)\text{ with distinct }{\lambda }_j$)

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 $T-{\lambda }_{j}I$. Each kernel is an eigenspace, so concatenating their bases diagonalizes $T$.