# On the spectra of $g$-circulant matrices and applications

## TL;DR

This paper investigates the spectral properties of g-circulant matrices, constructs nonnegative examples using number theory, and applies these findings to the Nonnegative Inverse Eigenvalue Problem, extending to block matrices.

## Contribution

It introduces new spectral conditions for g-circulant matrices and applies them to construct matrices with prescribed spectra, advancing understanding in matrix theory.

## Key findings

- Spectral conditions differ between g-circulant and circulant matrices.
- Constructed nonnegative g-circulant matrices with specific spectra.
- Extended results to block g-circulant matrices with orders multiple of a prime.

## Abstract

A g-circulant matrix of order n is defined as a matrix of order n where each row is a right cyclic shift in g-places to the preceding row. Using number theory, certain nonnegative g-circulant real matrices are constructed. In particular, it is shown that spectra with sufficient conditions so that it can be the spectrum of a real g-circulant matrix is not a spectrum with sufficient conditions so that it can be the spectrum of a real circulant matrix of the same order. The obtained results are applied to Nonnegative Inverse Eigenvalue Problem to construct nonnegative, g-circulant matrices with given appropriated spectrum. Moreover, nonnegative g-circulant matrices by blocks are also studied and in this case, their orders can be a multiple of a prime number.