Toeplitz nonnegative realization of spectra via companion matrices

Macarena Collao,Mario Salas,Ricardo L Soto

doi:10.1515/spma-2019-0017

Macarena Collao, Mario Salas + Show 1 more

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https://doi.org/10.1515/spma-2019-0017

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Journal: Special Matrices	Publication Date: Jan 1, 2019
Citations: 1	License type: CC BY 4.0

Affiliation: Universidad Católica del Norte

Abstract

Abstract The nonnegative inverse eigenvalue problem (NIEP) is the problem of finding conditions for the existence of an n × n entrywise nonnegative matrix A with prescribed spectrum Λ = {λ 1, . . ., λn}. If the problem has a solution, we say that Λ is realizable and that A is a realizing matrix. In this paper we consider the NIEP for a Toeplitz realizing matrix A, and as far as we know, this is the first work which addresses the Toeplitz nonnegative realization of spectra. We show that nonnegative companion matrices are similar to nonnegative Toeplitz ones. We note that, as a consequence, a realizable list Λ= {λ 1, . . ., λn} of complex numbers in the left-half plane, that is, with Re λi ≤ 0, i = 2, . . ., n, is in particular realizable by a Toeplitz matrix. Moreover, we show how to construct symmetric nonnegative block Toeplitz matrices with prescribed spectrum and we explore the universal realizability of lists, which are realizable by this kind of matrices. We also propose a Matlab Toeplitz routine to compute a Toeplitz solution matrix.

Highlights

The nonnegative inverse eigenvalue problem is the problem of nding necessary and su cient conditions for a list Λ = {λ, λ, . . . , λn} of complex numbers to be the spectrum of an n × n entrywise nonnegative matrix
We show how to construct symmetric nonnegative block Toeplitz matrices with prescribed spectrum and we explore the universal realizability of lists, which are realizable by this kind of matrices
If there exists a nonnegative matrix A with spectrum Λ, we say that Λ is realizable and that A is a realizing matrix

Summary

Introduction

In this work we show, under certain conditions, that a list of complex numbers is the spectrum of a Toeplitz nonnegative matrix, which is not necessarily circulant. In this paper we give su cient conditions for the existence and construction of a nonnegative Toeplitz matrix with prescribed spectrum. In Section , we give su cient conditions for the existence of a symmetric nonnegative block Toeplitz matrix with prescribed Jordan canonical form.

Results

Conclusion