Friedland, Shmuel A note on the nonzero spectra of irreducible matrices. (English) Zbl 1257.15006 Linear Multilinear Algebra 60, No. 11-12, 1235-1238 (2012). The author shows that given a multiset of nonzero complex numbers \(\Lambda\), it is a nonzero eigenvalue multiset of a nonnegative irreducible matrix if and only if it is a Frobenius set and some inequalities are satisfied in terms of elementary symmetric functions of the eigenvalues. This extends the theorem of Boyle-Handelman which deals with the nonnegative primitive case (cf. [M. Boyle and D. Handelman, Ann. Math. (2) 133, No. 2, 249–316 (1991; Zbl 0735.15005)]. The author also extends a theorem of K. H. Kim, N. S. Ormes and F. W. Roush, [“The spectra of nonnegative integer matrices via formal power series”, J. Am. Math. Soc. 13, No. 4, 773–806 (2000; Zbl 0968.15005)]. The extension is from nonnegative primitive integral matrices to nonnegative irreducible integral matrices. Reviewer: Tin Yau Tam (Auburn) Cited in 1 Document MSC: 15A18 Eigenvalues, singular values, and eigenvectors 15A29 Inverse problems in linear algebra 15A42 Inequalities involving eigenvalues and eigenvectors 15B36 Matrices of integers 15B48 Positive matrices and their generalizations; cones of matrices Keywords:nonnegative inverse eigenvalue problem; primitive matrices; Boyle-Handelman theorem; Kim-Ormes-Roush theorem; nonnegative irreducible integral matrices Citations:Zbl 0735.15005; Zbl 0968.15005 PDFBibTeX XMLCite \textit{S. Friedland}, Linear Multilinear Algebra 60, No. 11--12, 1235--1238 (2012; Zbl 1257.15006) Full Text: DOI arXiv Link References: [1] DOI: 10.2307/2944339 · Zbl 0735.15005 · doi:10.2307/2944339 [2] DOI: 10.1007/BF02760401 · Zbl 0407.15015 · doi:10.1007/BF02760401 [3] Gantmacher FR, The Theory of Matrices (1959) [4] Horn RA, Matrix Analysis (1988) [5] DOI: 10.1090/S0894-0347-00-00342-8 · Zbl 0968.15005 · doi:10.1090/S0894-0347-00-00342-8 [6] Laffey TJ, A constructive version of the Boyle–Handelman theorem on the spectra of nonnegative matrices [7] Laffey T, Electron. J. Linear Algebra 3 pp 119– (1998) · Zbl 0907.15013 · doi:10.13001/1081-3810.1018 [8] DOI: 10.1080/03081087808817226 · Zbl 0376.15006 · doi:10.1080/03081087808817226 [9] DOI: 10.1016/j.laa.2007.06.014 · Zbl 1136.15007 · doi:10.1016/j.laa.2007.06.014 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.