Abstract

A g-circulant matrix A, is defined as a matrix of order n where the elements of each row of A are identical to those of the previous row, but are moved g positions to the right and wrapped around. Using number theory, certain spectra of g-circulant real matrices are given explicitly. The obtained results are applied to Nonnegative Inverse Eigenvalue Problem to construct nonnegative, g-circulant matrices with given appropriated spectrum. Additionally, some g-circulant matrices are reconstructed from its main diagonal entries.

Highlights

  • A permutative matrix is a square matrix where each row is a permutation of its first row

  • If the list Σ is realizable by a nonnegative matrix A, we say that A realizes Σ or it is a realizing matrix for Σ

  • G = n − 1.We conclude that a g-circulant matrix of order n, with n prime is symmetric if and only if g ≡ n − 1(mod n)

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Summary

Introduction

A permutative matrix is a square matrix where each row is a permutation of its first row. The study of circulant matrices and their use in realizing lists of complex numbers has produced many results. Sufficient conditions in order that a given list can be taken as the spectrum of a nonnegative g-circulant matrix are given. G = n − 1.We conclude that a g-circulant matrix of order n, with n prime is symmetric if and only if g ≡ n − 1(mod n). It is clear that all the rows of a g−circulant matrix are distinct if and only if (n, g) = 1, see [4] and, this is the case that we study in this work In this case, the rows of the g-circulant matrix can be permuted in such a way to re-obtain a classical circulant matrix. The eigenvalues of a g-circulant matrix A are obtained in [26] by using the equality in (4)

Characterizing the spectra of certain g-circulant matrices
Reconstructing certain g-circulant matrices from its diagonal entries

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