Krylov subspace

Summary

The Krylov subspace of order $r$ generated by a square matrix $A$ and a vector $b$ is the linear span of $b, A b, \dots, A^{r-1} b$. Its elements are the vectors $p(A) b$ for polynomials $p$ of degree less than $r$. Many iterative methods for large linear systems and eigenvalue problems search for approximations in Krylov subspaces. These subspaces can be built with matrix-vector products alone.