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Existence and construction of nonnegative matrices with prescribed spectrum

โœ Scribed by Ricardo L. Soto

Publisher
Elsevier Science
Year
2003
Tongue
English
Weight
139 KB
Volume
369
Category
Article
ISSN
0024-3795

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โœฆ Synopsis

We consider the following inverse spectrum problem for nonnegative matrices: given a set of real numbers ฯƒ = {ฮป 1 , ฮป 2 , . . . , ฮป n }, find necessary and sufficient conditions for the existence of an n ร— n nonnegative matrix A with spectrum ฯƒ . In particular, by the use of a relevant theorem of Brauer we obtain new simple sufficient conditions for the problem to have a solution. Moreover, we can always construct a solution matrix, which is nonnegative generalized stochastic.

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## In this paper, for a prescribed real spectrum, using properties of the circulant matrices and of the symmetric persymmetric matrices, we derive a fast and stable algorithm to construct a symmetric nonnegative matrix which realizes the spectrum. The algorithm is based on the fast Fourier transfo