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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.

A Krylov subspace method seeks approximations in spaces generated by repeated matrix applications to a starting vector. For a linear system with starting point $x_0$ and residual $r_0=b-A{x}_0$, the affine search space is $x_0+K_k(A,r_0)$. Choosing an optimality or orthogonality condition within it gives different algorithms. The conjugate gradient method minimizes the energy norm of the error for a symmetric positive-definite matrix; other methods use residual minimization or different projections. The growth of the Krylov subspace can stop early when an invariant subspace is reached.

The Krylov subspace is $\boxed{K_k(A,b)=\mathrm{span}\{b,Ab,\dots ,A^{k-1}b\}}$. If $y\in K_k$, write $y=q(A)b$ with $\mathrm{deg}q\le k-1$. Since $b=Ax$,which has degree at most $k$ and $p(0)=1$. Conversely, if $x-y=p(A)x$ with these properties, then $q(t)=(1-p(t))/t$ is a polynomial of degree at most $k-1$ andThis is the polynomial description of the admissible errors in a Krylov subspace method.

The Krylov subspace is $K_k(A,r)=\mathrm{span}\{r,Ar,\dots ,A^{k-1}r\}$. Write $x_k-x_0=q(A)r_0$ with $\mathrm{deg}q\le k-1$. For the exact solution, $r_0=A(x-x_0)$, soThis polynomial has degree at most $k$ and $p(0)=1$. It is the error-polynomial representation for a Krylov subspace method.