The distance of an eigenvector to a Krylov subspace and the convergence of the Arnoldi method for eigenvalue problems

We study the distance of an eigenvector of a diagonalizable matrix A to the Krylov subspace generated from A and a given starting vector v. This distance is involved in studies of the convergence of the Arnoldi method for computing eigenvalues. Contrary to the previous studies on this problem, we provide closed-form expressions for this distance in terms of the eigenvalues and eigenvectors of A as well as the components of v in the eigenvector basis. The formulas simplify when the matrix A is normal. For A non-normal we derive upper and lower bounds that are simpler than the exact expressions. We also show how to generate starting vectors such that the distance to the Krylov subspace is equal to the worst possible case.