Rank-1 perturbations and the Lanczos method, inverse iteration, and Krylov subspaces

The heart of the Lanczos algorithm is the systematic generation of orthonormal bases of invariant subspaces of a perturbed matrix. The perturbations involved are special since they are always rank-1 and are the smallest possible in certain senses. These minimal perturbation properties are extended here to more general cases.Rank-1 perturbations are also shown to be closely connected to inverse iteration, and thus provide a novel explanation of the global convergence phenomenon of Rayleigh quotient iteration.Finally, we show that the restriction to a Krylov subspace of a matrix differs from the restriction of its inverse by a rank-1 matrix.