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A note on the nonzero spectra of irreducible matrices. (English) Zbl 1257.15006

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.

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
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[1] DOI: 10.2307/2944339 · Zbl 0735.15005 · doi:10.2307/2944339
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[5] DOI: 10.1090/S0894-0347-00-00342-8 · Zbl 0968.15005 · doi:10.1090/S0894-0347-00-00342-8
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