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Existence and construction of nonnegative matrices with complex spectrum. (English) Zbl 1031.15017

The paper is concerned with the following inverse spectrum problem: Given a family \(\sigma=(\lambda_1,\lambda_2,\ldots,\lambda_n)\) of complex numbers find necessary and sufficient conditions for the existence of a (real) nonnegative \((n\times n)\)-matrix \(A\) with spectrum \(\sigma\). It is based on a characterization due to W. Guo [ibid. 266, 261-270 (1997; Zbl 0903.15003)] which guarantees the existence of a solution matrix \(A\) provided that \(\lambda_1\) is a real number satisfying \(\lambda_1\geq 2n \max_{2\leq j\leq n}|\lambda_j|\). From this result the authors derive necessary and sufficient conditions for a certain type of \(\sigma\)’s to be the spectrum of a circulant nonnegative matrix. Also, methods how to find explicit solutions are presented.

MSC:

15A29 Inverse problems in linear algebra
15A18 Eigenvalues, singular values, and eigenvectors
15B48 Positive matrices and their generalizations; cones of matrices
15B51 Stochastic matrices

Citations:

Zbl 0903.15003
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References:

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