# Realizable lists via the spectra of structured matrices

## TL;DR

This paper investigates the spectra of permutative matrices, especially those partitioned into symmetric blocks, providing conditions for lists to be eigenvalues of such matrices and constructing examples.

## Contribution

It introduces spectral results for block-structured permutative matrices and offers new conditions and constructions for their eigenvalue lists.

## Key findings

- Spectral results for matrices partitioned into 2-by-2 symmetric blocks.
- Sufficient conditions for a list to be eigenvalues of nonnegative permutative matrices.
- Construction of permutative matrices with prescribed spectra.

## Abstract

A square matrix of order $n$ with $n\geq 2$ is called a \textit{permutative matrix} or permutative when all its rows (up to the first one) are permutations of precisely its first row. In this paper, the spectra of a class of permutative matrices are studied. In particular, spectral results for matrices partitioned into $2$-by-$2$ symmetric blocks are presented and, using these results sufficient conditions on a given list to be the list of eigenvalues of a nonnegative permutative matrix are obtained and the corresponding permutative matrices are constructed. Guo perturbations on given lists are exhibited.