Eigenvalues of non-Hermitian matrices: A dynamical and an iterative approach—Application to a truncated Swanson model

Abstract

We propose two different strategies to find eigenvalues and eigenvectors of a given, not necessarily Hermitian, matrix (Formula presented.). Our methods apply also to the case of complex eigenvalues, making the strategies interesting for applications to physics and to pseudo-Hermitian quantum mechanics in particular. We first consider a dynamical approach, based on a pair of ordinary differential equations defined in terms of the matrix (Formula presented.) and of its adjoint (Formula presented.). Then, we consider an extension of the so-called power method, for which we prove a fixed point theorem for (Formula presented.) useful in the determination of the eigenvalues of (Formula presented.) and (Formula presented.). The two strategies are applied to some explicit problems. In particular, we compute the eigenvalues and the eigenvectors of the matrix arising from a recently proposed quantum mechanical system, the truncated Swanson model, and we check some asymptotic features of the Hessenberg matrix.