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

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 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. 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 truncated Swanson model, and we check some asymptotic features of the Hessenberg matrix.