Eigenvalues of non‐Hermitian matrices: A dynamical and an iterative approach—Application to a truncated Swanson model
Fabio Bagarello, Francesco Gargano · Mathematical Methods in the Applied Sciences · 2020
We propose two different strategies to find eigenvalues and eigenvectors of a given, not necessarily Hermitian, matrix . 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 and of its adjoint . Then, we consider an extension of the so‐called power method, for which we prove a fixed point theorem for useful in the determination of the eigenvalues of and . 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.