Normal nonnegative realization of spectra

The nonnegative inverse eigenvalue problem is the problem of finding necessary and sufficient conditions for the existence of an entrywise nonnegative matrix A with prescribed spectrum. This problem remains open for . If the matrix A is required to be normal, the problem will be called the normal no...

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Published inLinear & multilinear algebra Vol. 63; no. 6; pp. 1204 - 1215
Main Authors Julio, Ana I., Manzaneda, Cristina B., Soto, Ricardo L.
Format Journal Article
LanguageEnglish
Published Abingdon Taylor & Francis 03.06.2015
Taylor & Francis Ltd
Subjects
Algebra
Complex numbers
Construction
Economic models
Eigenvalues
Inverse
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normal nonnegative inverse eigenvalue problem
Spectra
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Summary:The nonnegative inverse eigenvalue problem is the problem of finding necessary and sufficient conditions for the existence of an entrywise nonnegative matrix A with prescribed spectrum. This problem remains open for . If the matrix A is required to be normal, the problem will be called the normal nonnegative inverse eigenvalue problem (NNIEP). Sufficient conditions for a list of complex numbers to be the spectrum of a normal nonnegative matrix were obtained by Xu [Linear Multilinear Algebra. 1993;34:353-364]. In this paper, we give a normal version of a rank-r perturbation result due to Rado and published by Perfect [Duke Math. J. 1955;22:305-311], which allow us to obtain new sufficient conditions for the NNIEP to have a solution. These new conditions significantly improve Xu's conditions. We also apply our results to construct nonnegative matrices with arbitrarily prescribed elementary divisors.
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ISSN:0308-1087
1563-5139
DOI:10.1080/03081087.2014.924513