Realizable Lists on a Class of Nonnegative Matrices

TL;DR

This paper investigates conditions under which complex eigenvalue lists can be realized by nonnegative permutative matrices, providing spectral results, realizability regions, and insights into NIEP and PNIEP for matrices with specific complex entries.

Contribution

It offers new sufficient conditions for realizing complex eigenvalue lists with nonnegative permutative matrices, especially for matrices with certain real or imaginary parts.

Findings

Derived spectral conditions for permutative matrices.

Identified realizability regions for nonnegative permutative matrices.

Analyzed NIEP and PNIEP with complex eigenvalues having no zero imaginary part.

Abstract

A square matrix of order n with $n\geq 2$ is called permutative matrix when all its rows (up to the frst one) are permutations of precisely its frst row. In this paper recalling spectral results for partitioned into $2$-by-$2$ symmetric blocks matrices sufcient conditions on a given complex list to be the list of the eigenvalues of a nonnegative permutative matrix are given. In particular, we study NIEP and PNIEP when some complex elements into the considered lists have no zero imaginary part. Realizability regions for nonnegative permutative matrices are obtained.