Abstract
One consequence of the Perron–Frobenius Theorem on indecomposable positive matrices is that whenever an \(n\times n\) matrix A dominates a non-singular positive matrix, there is an integer k dividing n such that, after a permutation of basis, A is block-monomial with \(k\times k\) blocks. Furthermore, for suitably large exponents, the nonzero blocks of \(A^m\) are strictly positive. We present an extension of this result for indecomposable semigroups of positive matrices.
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Acknowledgments
L. Livshits was supported by the Colby College Natural Science Division Grant. G. MacDonald and H. Radjavi acknowledge the support of NSERC Canada.
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Livshits, L., MacDonald, G. & Radjavi, H. A Perron-Frobenius-type Theorem for Positive Matrix Semigroups. Positivity 21, 61–72 (2017). https://doi.org/10.1007/s11117-016-0403-7
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DOI: https://doi.org/10.1007/s11117-016-0403-7