Eigenvalue and pseudospectrum processes generated by nonnormal Toeplitz matrices with rank 1 perturbations

We introduce two kinds of matrix-valued dynamical processes generated by nonnormal Toeplitz matrices with the additive rank 1 perturbations [Formula: see text], where [Formula: see text] and [Formula: see text] is the all-ones matrix. For each process, first we report the complicated motion of the numerically obtained eigenvalues. Then we derive the specific equation which determines the motion of non-zero simple eigenvalues and clarifies the time-dependence of degeneracy of the zero-eigenvalue [Formula: see text]. Comparison with the solutions of this equation, it is concluded that the numerically observed non-zero eigenvalues distributing around [Formula: see text] are the exact eigenvalues not of the original system, but of the system perturbed by uncontrolled rounding errors of computer. The complex domain in which the eigenvalues of randomly perturbed system are distributed is identified with the pseudospectrum including [Formula: see text] of the original system with [Formula: see text]. We characterize the pseudospectrum processes using the symbol curves of the corresponding nonnormal Toeplitz operators without [Formula: see text]. We report new phenomena in our second model such that at each time the outermost closed simple curve cut out from the symbol curve is realized as the exact eigenvalues, but the inner part of symbol curve is reduced in size and embedded in the pseudospectrum including [Formula: see text]. Such separation of exact simple eigenvalues and a degenerated eigenvalue associated with pseudospectrum will be meaningful for numerical analysis, since the former is stable and robust, but the latter is highly sensitive and unstable with respective to perturbations. The present study will be related to the pseudospectra approaches to non-Hermitian systems developed in quantum physics.