Annihilating polynomial ($p(A)=0$)

An annihilating polynomial of a linear operator is a nonzero polynomial $p$ for which the displayed operator is zero. The minimal polynomial is its unique monic annihilating polynomial of least degree and divides every other annihilating polynomial. For a diagonalizable operator with distinct eigenvalues ${\lambda }_1,\dots ,{\lambda }_s$, the product ${\prod }_j(t-{\lambda }_j)$ annihilates it. For an individual vector it suffices to annihilate the eigenvalues contributing to that vector; this gives exact finite-step termination bounds for a Krylov subspace method.

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.