# A Perron-Frobenius-type Theorem for Positive Matrix Semigroups

A Perron-Frobenius-type Theorem for Positive Matrix Semigroups One consequence of the Perron–Frobenius Theorem on indecomposable positive matrices is that whenever an $$n\times n$$ n × 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 $$k\times k$$ k × k blocks. Furthermore, for suitably large exponents, the nonzero blocks of $$A^m$$ A m are strictly positive. We present an extension of this result for indecomposable semigroups of positive matrices. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# A Perron-Frobenius-type Theorem for Positive Matrix Semigroups

Positivity, Volume 21 (1) – Mar 3, 2016
12 pages

/lp/springer_journal/a-perron-frobenius-type-theorem-for-positive-matrix-semigroups-2M0WuS560g
Publisher
Springer Journals
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-016-0403-7
Publisher site
See Article on Publisher Site

### Abstract

One consequence of the Perron–Frobenius Theorem on indecomposable positive matrices is that whenever an $$n\times n$$ n × 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 $$k\times k$$ k × k blocks. Furthermore, for suitably large exponents, the nonzero blocks of $$A^m$$ A m are strictly positive. We present an extension of this result for indecomposable semigroups of positive matrices.

### Journal

PositivitySpringer Journals

Published: Mar 3, 2016

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