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 $A$. 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 {\em dynamical} approach, based on a pair of ordinary differential equations defined in terms of the matrix $A$ and of its adjoint $A^{†}$. Then we consider an extension of the so-called power method, for which we prove a fixed point theorem for $A\ne A^{†}$ useful in the determination of the eigenvalues of $A$ and $A^{†}$. 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 {\em truncated Swanson model}, and we check some asymptotic features of the Hessenberg matrix.