On semigroups of matrices with nonnegative diagonals

arXiv (Cornell University), 2012

Linear Algebra and its Applications, 2019

We call a function of k variables reducing for a multiplicative semigroup S ⊆ M n (C), if the semigroup is reducible whenever f sends S to zero. For continuous reducing functions on groups of unitary matrices we prove that such a group is reducible whenever f is uniformly small enough on the group. Extensions of this result for certain types of semigroups are also given. Note that our results contain and extend many known results of the kind. In particular, it is shown that a group of complex unitary matrices is reducible if, for fixed indices i and j, its (i, j) entries are uniformly small.

Linear Algebra and its Applications

Multiplicative matrix semigroups with constant spectral radius (c.s.r.) are studied and applied to several problems of algebra, combinatorics, functional equations, and dynamical systems. We show that all such semigroups are characterized by means of irreducible ones. Each irreducible c.s.r. semigroup defines walks on Euclidean sphere, all its nonsingular elements are similar (in the same basis) to orthogonal. We classify all nonnegative c.s.r. semigroups and arbitrary low-dimensional semigroups. For higher dimensions, we describe five classes and leave an open problem on completeness of that list. The problem of algorithmic recognition of c.s.r. property is proved to be polynomially solvable for irreducible semigroups and undecidable for reducible ones.

Semigroup Forum, 1971

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Linear Algebra and Its Applications, 2007

We prove that any pure regular band of matrices admits a simultaneous LU decomposition in the standard form. In case that such a band forms a double-band called a skew lattice, we obtain the standard form without the assumption of purity.

Linear Algebra and its Applications, 2006

Characterizations are given for automorphisms of semigroups of nonnegative matrices including doubly stochastic matrices, row (column) stochastic matrices, positive matrices, and nonnegative monomial matrices. The proofs utilize the structure of the automorphisms of the symmetric group (realized as the group of permutation matrices) and alternating group . Furthermore, for each of the above (semi)groups of matrices, a larger (semi)group of matrices is obtained by relaxing the nonnegativity assumption. Characterizations are also obtained for the automorphisms on the larger (semi)groups and their subgroups (subsemigroups) as well.

Linear Algebra and its Applications, 2011

Unlike factorization theory of commutative semigroups which are well-studied, very little literature exists describing factorization properties in noncommutative semigroups. Perhaps the most ubiquitous noncommutative semigroups are semigroups of square matrices and this article investigates the factorization properties within certain subsemigroups of M n (Z), the semigroup of n × n matrices with integer entries. Certain important invariants are calculated to give a sense of how unique or non-unique factorization is in each of these semigroups.

Canadian Mathematical Bulletin, 2012

We use the kernel category to give a finiteness condition for semigroups. As a consequence we provide yet another proof that finitely generated periodic semigroups of matrices are finite.

Linear Algebra and its Applications, 2003

In this paper we give necessary and sufficient conditions for a matrix in Jordan canonical form to be similar to an eventually nonnegative matrix whose irreducible diagonal blocks satisfy the conditions identified by Zaslavsky and Tam, and whose subdiagonal blocks (with respect to its Frobenius normal form) are nonnegative. These matrices are referred to as seminonnegative matrices, and we show that they exhibit many of the same combinatorial spectral properties as nonnegative matrices. This paper extends the work on Jordan forms of irreducible eventually nonnegative matrices to the reducible case.

Ars Mathematica Contemporanea

Let F be a field. We classify multiplicative maps from M n (F) to M (n k) (F) which annihilate a zero matrix and map rank-k matrix into a rank-one matrix.