A comparison of nonnegative real ranks and their preservers

This paper concerns three notions of rank of matrices over semirings; real rank, semiring rank and column rank. These three rank functions are the same over subfields of reals but differ for matrices over subsemirings of nonnegative reals. We investigate the largest values of r for which the real rank and semiring rank, real rank and column rank of all m×n matrices over a given semiring are both r, respectively. We also characterize the linear operators which preserve the column rank of matrices over certain subsemirings of the nonnegative reals.     

A comparison of nonnegative real ranks and their preservers

This paper concerns three notions of rank of matrices over semirings; real rank, semiring rank and column rank. These three rank functions are the same over subfields of reals but differ for matrices over subsemirings of nonnegative reals. We investigate the largest values of r for which the real rank and semiring rank, real rank and column rank of all m×n matrices over a given semiring are both r, respectively. We also characterize the linear operators which preserve the column rank of matrices over certain subsemirings of the nonnegative reals.     

Bijective linear rank preservers for spaces of matrices over antinegative semirings

We classify the bijective linear operators on spaces of matrices over antinegative commutative semirings with no zero divisors which preserve certain rank functions such as the symmetric rank, the factor rank and the tropical rank. We also classify the bijective linear operators on spaces of matrices over the max-plus semiring which preserve the Gondran-Minoux row rank or the Gondran-Minoux column rank.     

Generalized fuzzy matrices

We give a systematic development of fuzzy matrix theory. Many of our results generalize to matrices over the two element Boolean algebra, over the nonnegative real numbers, over the nonnegative integers, and over the semirings, and we present these generalizations. Our first main result is that while spaces of fuzzy vectors do not have a unique basis in general they have a unique standard basis, and the cardinality of any two bases are equal. Thus concepts of row and column basis, row and column rank can be defined for fuzzy matrices. Then we study Green's equivalence classes of fuzzy matrices. New we give criteria for a fuzzy matrix to be regular and prove that the row and column rank of any regular fuzzy matrix are equal. Various inverses are also studied. In the next section, we obtain bounds for the index and period of a fuzzy matrix.     

Extreme preservers of maximal column rank inequalities of matrix sums over semirings

We characterize linear operators that preserve sets of matrix ordered pairs which satisfy extreme properties with respect to maximal column rank inequalities of matrix sums over semirings. This work was supported by the research grant of the Cheju National University in 2007.     

On the coincidence of the factor and Gondran–Minoux rank functions of matrices over a semiring

We consider the rank functions of matrices over semirings, functions that generalize the classical notion of the rank of a matrix over a field. We study semirings over which the factor and Gondran–Minoux ranks of any matrix coincide. It is shown that every semiring satisfying that condition is a subsemiring of a field. We provide an example of an integral domain over which the factor and Gondran–Minoux ranks are different.     

Inequalities for Gondran-Minoux rank and idempotent semirings

The rank-sum, rank-product, and rank-union inequalities for Gondran-Minoux rank of matrices over idempotent semirings are considered. We prove these inequalities for matrices over quasi-selective semirings without zero divisors, which include matrices over the max-plus semiring. Moreover, it is shown that the inequalities provide the linear algebraic characterization for the class of quasi-selective semirings. Namely, it is proven that the inequalities hold for matrices over an idempotent semiring S without zero divisors if and only if S is quasi-selective. For any idempotent semiring which is not quasi-selective it is shown that the rank-sum, rank-product, and rank-union inequalities do not hold in general. Also, we provide an example of a selective semiring with zero divisors such that the rank-sum, rank-product, and rank-union inequalities do not hold in general.     

Zero-term rank preservers

We obtain characterizations of those linear operators that preserve zero-term rank on the m×n matrices over antinegative semirings. That is, a linear operator T preserves zero-term rank if and only if it has the form T(X)=P(B∘X)Q, where P, Q are permutation matrices and B∘X is the Schur product with B whose entries are all nonzero and not zero-divisors.     

Column Ranks and Their Preservers of Matrices Over Max Algebra

The column rank of an m by n matrix A over max algebra is the weak dimension of the column space of A. We compare the column rank with rank of matrices over max algebra. We also characterize the linear operators which preserve the column rank of matrices over max algebra.     

Factor and term ranks for matrix union over semirings

We consider the arithmetic properties of the factor-rank and term-rank functions for matrices over semirings. In particular, we investigate the sets of matrices that satisfy the extremes of inequalities for these rank functions of matrix union. The classification of the linear transformations that keep these sets invariant is obtained. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 3, pp. 175–197, 2003.     

Column Ranks and Their Preservers of Matrices Over Max Algebra

The column rank of an m by n matrix A over max algebra is the weak dimension of the column space of A. We compare the column rank with rank of matrices over max algebra. We also characterize the linear operators which preserve the column rank of matrices over max algebra.     

<Emphasis Type="Italic">e</Emphasis>-Invertible Matrices Over Commutative Semirings

In this article, the e-invertible matrices over commutative semirings are studied. Some properties and equivalent characterizations of the e-invertible matrices are given. Also, the interrelationships between invertible matrices and e-invertible matrices over commutative semirings are discussed. The main results obtained in this article generalize and enrich the corresponding results about invertible matrices over commutative semirings.     

Linear operators strongly preserving r-cyclic matrices over semirings

We say that A is an r-cyclic matrix if A^r=I. We investigate the structure of linear operators on matrices over antinegative semirings that map r-cyclic matrices to r-cyclic matrices and non r-cyclic matrices to non r-cyclic matrices.     

Linear operators that preserve maximal column ranks of nonnegative integer matrices

The maximal column rank of an by matrix over a semiring is the maximal number of the columns of which are linearly independent. We characterize the linear operators which preserve the maximal column ranks of nonnegative integer matrices.

     

Two concepts of singularity for matrices over semirings

The two common concepts of singularity for matrices over semirings are being studied since the 1970’s and arise from natural generalizations of the determinant and linear dependence. They were introduced in the context of schedule algebras by Gondran and Minoux, who proved later that the concepts discussed are equivalent over any selective invertible semiring. We present an approach that uses a generalization of power series arithmetic and, in particular, allows to derive a short proof for the theorem of Gondran and Minoux. Our main result is a complete concise characterization of semirings over which the two concepts of singularity are equivalent.     

Invertible Matrices over Finite Additively Idempotent Semirings

We investigate invertible matrices over finite additively idempotent semirings. The main result provides a criterion for the invertibility of such matrices. We also give a construction of the inverse matrix and a formula for the number of invertible matrices.     

半环上矩阵的秩和坡上矩阵可逆的条件

研究了交换半环上矩阵的秩和坡上矩阵的可逆条件.利用Beasley的引理以及不变式,获得了交换半环上正则矩阵的行秩、列秩与Schein秩三者相等,以及坡上矩阵可逆的充要条件.推广模糊代数和分配格上矩阵的结果.     

A geometric treatment of generalized inverses and semigroups of nonnegative matrices

The purpose of this paper is to provide a unified treatment from the geometric viewpoint of the following closely related aspects of nonnegative matrices: nonnegative matrices with nonnegative generalized inverses of various kinds; nonnegative rank factorization; regular elements, Green's relations, and maximal subgroups of the semigroups of nonnegative matrices, stochastic matrices, column stochastic matrices, and doubly stochastic matrices.