Borobia, Alberto On the nonnegative eigenvalue problem. (English) Zbl 0831.15014 Linear Algebra Appl. 223-224, 131-140 (1995). Author’s summary: \(\{1, \lambda_1, \ldots, \lambda_n\}\) is the spectrum of a stochastic matrix of order \(n + 1\) if and only if there exists some real matrix \(B\) of order \(n\) with spectrum \(\{\lambda_1, \ldots, \lambda_n\}\) and some \(n\)-simplex \(S \subset \mathbb{R}^n\) containing the origin such that \(BS \subset S\). This result appears implicitly in P. G. Ciarlet’s work [ibid. 1, 139-152 (1968; Zbl 0167.031)], and it provides us with a geometrical tool to obtain sufficient conditions for the nonnegative eigenvalue problem. We employ it here to generalize the sufficient conditions given by R. B. Kellogg [ibid. 4, 191-204 (1971; Zbl 0215.375)]. Reviewer: A.Lal (Allahabad) Cited in 2 ReviewsCited in 39 Documents MSC: 15B51 Stochastic matrices 15A18 Eigenvalues, singular values, and eigenvectors Keywords:spectrum; stochastic matrix; nonnegative eigenvalue problem Citations:Zbl 0167.031; Zbl 0215.375 PDFBibTeX XMLCite \textit{A. Borobia}, Linear Algebra Appl. 223--224, 131--140 (1995; Zbl 0831.15014) Full Text: DOI References: [1] Ciarlet, P. G., Some results in the theory of nonnegative matrices, Linear Algebra Appl., 1, 139-152 (1968) · Zbl 0167.03101 [2] Fiedler, M., Eigenvalues of nonnegative symmetric matrices, Linear Algebra Appl., 9, 119-142 (1974) · Zbl 0322.15015 [3] Johnson, C. R., Row stochastic matrices similar to doubly stochastic matrices, Linear and Multilinear Algebra, 10, 113-130 (1981) · Zbl 0455.15019 [4] Kellogg, R. B., Matrices similar to a positive or essentially positive matrix, Linear Algebra Appl., 4, 191-204 (1971) · Zbl 0215.37504 [5] Perfect, H., On positive stochastic matrices with real characteristic roots, (Proc. Cambridge Philos. Soc., 48 (1952)), 271-276 · Zbl 0046.35003 [6] Salzmann, F., A note on eigenvalues of nonnegative matrices, Linear Algebra Appl., 5, 329-338 (1972) · Zbl 0255.15009 [7] Suleimanova, H. R., Stochastic matrices with real characteristic values, Dokl. Akad. Nauk SSSR, 66, 343-345 (1949), (in Russian) · Zbl 0035.20903 [8] Oliveira, G., Sobre matrices estocásticas e duplamente estocásticas, Rev. Fac. Ciencias Univ. Coimbra, XLI (1968) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.