Block matrices and Guo's Index for block circulant matrices with circulant blocks

TL;DR

This paper investigates the spectral properties of block circulant matrices with circulant blocks, providing new conditions and the Guo's index related to the Nonnegative Inverse Eigenvalue Problem.

Contribution

It introduces spectral results for block circulant matrices with circulant blocks and derives the Guo's index, offering necessary and sufficient conditions for spectrum realizability.

Findings

Guo's index for block circulant matrices is explicitly determined.

Necessary and sufficient conditions for the NIEP are established.

Spectral realizability criteria are provided for nonnegative block matrices.

Abstract

In this paper we deal with circulant and partitioned into $n$-by-$n$ circulant blocks matrices and introduce spectral results concerning this class of matrices. The problem of finding lists of complex numbers corresponding to a set of eigenvalues of a nonnegative block matrix with circulant blocks is treated. Along the paper we call realizable list if its elements are the eigenvalues of a nonnegative matrix. The Guo's index $\lambda_0$ of a realizable list is the minimum spectral radius such that the list (up to the initial spectral radius) together with $\lambda_0$ is realizable. The Guo's index of block circulant matrices with circulant blocks is obtained, and in consequence, necessary and sufficient conditions concerning the NIEP, Nonnegative Inverse Eigenvalue Problem, for the realizability of some spectra are given.