Primes in the semigroup of non-negative matrices

A matrix A in the semigroup N_n of non-negative n×nmatrices is prime if A is not monomial and A=BC,BCεN_n implies that either B or C is monomial. One necessary and another sufficient condition are given for a matrix in N_n to be prime. It is proved that every prime in N_n is completely decomposable.     

Primes in the semigroup of Boolean matrices

Let A, B, C be n×n matrices of zeros and ones. Using Boolean addition and multiplication, we say that A is prime if it is not a permutation matrix and if A=BC implies that B or C must be a permutation matrix. Conditions sufficient for a matrix to be prime are provided, and a characterization of primes in terms of a nation of rank is given. Finally, an order property of primes is used to obtain a result on prime factors.     

Decomposably non-negative matrices

Let A be an mn- by - mn symmetric matrix. Partition A into m²n - by - n blocks and suppose that each of these blocks is also symmetric. Suppose that for every decomposable (rank one) tensor ν ⊗ w, we have (ν ⊗ w)^t A(ν otimes; w) ≥ 0. Here, ν is a column m-tuple and w is a column n-tuple. We study the maximum number of negative eigenvalues such a matrix can have, as well as obtaining alternative characterizations of such matrices.     

An irreducible semigroup of non-negative square-zero operators

We construct an irreducible multiplicative semigroup of non-negative square-zero operators acting onL ^p [0,1), for 1p<.The main idea for this paper was developed at the 2nd Linear Algebra Workshop at Bled, Slovenia, in June 1999.The work of the three Slovenian authors was supported by the Research Ministry of Slovenia.This author's work was supported by a Division grant from Colby College.     

On the eigenvalues of non-negative Jacobi matrices

The spectrum σ of a non-negative Jacobi matrix J is characterized. If J is also required to be irreducible, further conditions on σ are needed, some of which are explored.     

An inequality for non-negative matrices

Let A be an n × n matrix with non-negative entries and no entry in (0, 1). We prove that there exist integers r, s with 0 r s 2ⁿ such that A^r A^s. We prove that 2ⁿ cannot be replaced with e√n log n. We also give an application to the theory of formal languages.     

Bounded indecomposable semigroups of non-negative matrices

A semigroup \({\mathfrak{S}}\) of non-negative n × n matrices is indecomposable if for every pair i, j ≤ n there exists \({S\in\mathfrak{S}}\) such that (S) _ij ≠ 0. We show that if there is a pair k, l such that \({\{(S)_{kl} : S\in\mathfrak{S}\}}\) is bounded then, after a simultaneous diagonal similarity, all the entries are in [0, 1]. We also provide quantitative versions of this result, as well as extensions to infinite-dimensional cases.     

Perturbing non-real eigenvalues of non-negative real matrices

Let σ=(ρ,b+ic,b-ic,λ₄,…,λ_n) be the spectrum of an entry non-negative matrix and t?0. Laffey [T. J. Laffey, Perturbing non-real eigenvalues of nonnegative real matrices, Electron. J. Linear Algebra 12 (2005) 73-76] has shown that σ=(ρ+2t,b-t+ic,b-t-ic,λ₄,…,λ_n) is also the spectrum of some nonnegative matrix. Laffey (2005) has used a rank one perturbation for small t and then used a compactness argument to extend the result to all nonnegative t. In this paper, a rank two perturbation is used to deduce an explicit and constructive proof for all t?0.     

三阶单位上三角非负矩阵

根据矩阵分解性质及原子间的乘积关系,首先探究了单位上三角非负矩阵的原子因式分解性质,然后给出了任意单位上三角非负矩阵A的最小原子因式分解长度l(A)的计算公式,并得到了A具有最小的原子因式分解长度的其中一种分解,从而完善了相关文献中的结果.     

Compactness of eigenvectors for certain families of non-negative matrices

Conditions are given which guarantee that the normalized left eigenvectors of certain classes of non-negative matrices of unbounded dimension, interpreted as functions of [0, 1], form precompact sets in L₁. These eigenvectors are uniformly bounded above and in some cases uniformly bounded below.     

Automorphisms of the semigroup of invertible matrices with nonnegative elements

In this paper, we prove that every automorphism of the semigroup of invertible matrices with nonnegative elements over a linearly ordered associative ring on some specially defined subgroup coincides with the composition of an inner automorphism of the semigroup, an order-preserving automorphism of the ring, and a central homothety. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 2, pp. 3–23, 2005.     

Elementary equivalence of the semigroup of invertible matrices with nonnegative elements

In this paper, we prove that the semigroups of invertible matrices with nonnegative elements over linearly ordered associative rings are elementarily equivalent if and only if the matrices have the same dimension and the rings are elementarily equivalent as ordered rings. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 2, pp. 39–53, 2006.