Inverse Iteration: An Alternative Form

Rayleigh quotient inverse iteration (RQI) is examined from the standpoint of incremental vectors. By choosing such a vector orthogonal to the eigenvector iterate at the mth step, χ_m, a system of linear equations is derived; solving the system is shown to be equivalent to RQI. The residual norm converges to zero for Hermitian matrices, and the coefficient matrix does not in general converge to a singular matrix as it does in RQI. Furthermore, the procedure described produces successive vector iterates whose norms do not in general become large (when m→∞) as they do with RQI.