= Eigenspace decomposition from a square-free annihilating polynomial
{title2=$\prod_j(T-\mu_jI)=0\Longrightarrow V=\bigoplus_j\ker(T-\mu_jI)$}
For distinct <scalars> $\mu_j$, the <Lagrange interpolation polynomials> $\ell_j(t)=\prod_{k\ne j}(t-\mu_k)/(\mu_j-\mu_k)$ sum to one. Their values at $T$ have images in the respective <eigenspaces>, since $(t-\mu_j)\ell_j$ is a multiple of the <annihilating polynomial>. Hence every <vector> is the sum of its $\ell_j(T)$ images. On the $i$th <eigenspace>, $\ell_j(T)$ is multiplication by $\delta_{ij}$, which proves directness. This gives the decomposition without invoking the <primary decomposition theorem>. Under this decomposition a <linear operator> commutes with $T$ exactly when it preserves every <eigenspace>.
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