# Centrosymmetric nonnegative realization of spectra

## TL;DR

This paper characterizes which spectra can be realized by centrosymmetric nonnegative matrices, providing conditions and specific cases where realizability is guaranteed, especially for real spectra and small matrix sizes.

## Contribution

It introduces a characterization of spectra realizable by centrosymmetric nonnegative matrices, including new sufficient conditions and special cases for real spectra.

## Key findings

- Lists of nonnegative real numbers are realizable by centrosymmetric matrices.
- Lists of Suleimanova type (except one case) are realizable.
- For n=4, all realizable real spectra are centrosymmetric realizable.

## Abstract

A list $\Lambda =\{\lambda _{1},\lambda _{2},\ldots ,\lambda _{n}\}$ of complex numbers is said to be realizable if it is the spectrum of an entrywise nonnegative matrix. In this paper we intent to characterize those lists of complex numbers, which are realizable by a centrosymmetric nonnegative matrix. In particular, we show that lists of nonnegative real numbers, and lists of complex numbers of Suleimanova type (except in one particular case), are always the spectrum of some centrosymmetric nonnegative matrix. For the general lists we give sufficient conditions via a perturbation result. We also show that for $n=4,$ every realizable list of real numbers is also realizable by a nonnegative centrosymmetric matrix.