Non-negative persymmetric realizability of certain classes of spectra

TL;DR

This paper investigates the conditions under which certain spectra can be realized by non-negative persymmetric matrices, resolving an open problem and establishing equivalences with classical inverse eigenvalue problems.

Contribution

It resolves an open problem in the non-negative persymmetric inverse eigenvalue problem and establishes equivalence with the classical problem for trace-zero spectra.

Findings

Resolved the open problem in PNIEP.

Established equivalence between PNIEP and NIEP for trace-zero spectra.

Derived new sufficient conditions for spectra realizability with non-negative persymmetric matrices.

Abstract

Identifying the collection of scalars that represent a non-negative matrix's eigenvalues is known as the non-negative inverse eigenvalue problem (NIEP). Conditions for the existence of a non-negative matrix with a certain spectrum are examined in this work. The classical NIEP restricted to non-negative matrices having a persymmetric structure is the persymmetric non-negative inverse eigenvalue problem (PNIEP). We resolve the open problem stated in \cite{Ana}, and furthermore, equivalence of the PNIEP and NIEP is established for trace-zero spectra of five complex numbers. Also we obtain new sufficient conditions for the realizability of certain classes of spectra with non-negative persymmetric matrix realization. Based on generic structural features of persymmetric matrices and their characteristic polynomials, our method is constructive in nature. The effects of perturbations on imaginary part of complex eigenvalues are also analyzed and some perturbation results are derived.