On the permissible arrangements of Ritz values for normal matrices in the complex plane

This article discusses Ritz and harmonic Ritz values computed from a Krylov subspace, generated by a normal matrix. We give necessary and sufficient conditions for a given tuple of complex numbers to represent Ritz values: the existence of a positive solution to a certain linear system with a Cauchy matrix. Using this characterization, we prove several localization results for the Ritz values of normal matrices and discuss consequences for the convergence of the restarted Arnoldi algorithm. Similar results are shown for the harmonic case as well.