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BY 4.0 license Open Access Published by De Gruyter Open Access December 13, 2019

Toeplitz nonnegative realization of spectra via companion matrices

  • Macarena Collao , Mario Salas and Ricardo L. Soto EMAIL logo
From the journal Special Matrices


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 = 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= 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.

MSC 2010: 15A29; 15A18


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Received: 2019-07-25
Accepted: 2019-11-26
Published Online: 2019-12-13

© 2019 Macarena Collao et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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