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2012, Linear Algebra and its Applications
The term rank of an n × n matrix A is the least number of lines (rows or columns) needed to include all the nonzero entries in A.
Linear Algebra and its Applications, 2013
This paper concerns three notions of matrix functions over semirings; term rank, reduced term rank and star cover number. We compare these matrix functions and we study those linear operators that preserve these matrix functions of n × n symmetric matrices with zero diagonal. In particular, we obtain important characterizations of term-rank (reduced term rank, star cover number) preservers with respect to a term rank (a reduced term rank, a star cover number, respectively) and a special matrix.
Linear Algebra and its Applications, 2012
The term rank of a matrix A over a semiring S is the least number of lines (rows or columns) needed to include all the nonzero entries in A. In this paper, we study linear operators that preserve term ranks of matrices over S. In particular, we show that a linear operator T on matrix space over S preserves term rank if and only if T preserves term ranks 1 and α(≥2) if and only if T preserves two consecutive term ranks in a restricted condition. Other characterizations of termrank preservers are also given.
2012
The term rank of a matrix A is the least number of lines (rows or columns) needed to include all the nonzero entries in A, and is a well-known upper bound for many standard and non-standard matrix ranks, and is one of the most important combinatorially. In this paper, we obtain a characterization of linear operators that preserve term ranks of matrices over antinegative semirings. That is, we show that a linear operator T on a matrix space over antinegative semirings preserves term rank if and only if T preserves any two term ranks k and l if and only if T strongly preserves any one term rank k. 2010 Mathematics Subject Classiflcation : 15A86, 15A03 and 15A04.
Kyungpook mathematical journal, 2014
Let M(S) denote the set of all m×n matrices over a semiring S. For A ∈ M(S), zero-term rank of A is the minimal number of lines (rows or columns) needed to cover all zero entries in A. In [5], the authors obtained that a linear operator on M(S) preserves zero-term rank if and only if it preserves zero-term ranks 0 and 1. In this paper, we obtain new characterizations of linear operators on M(S) that preserve zero-term rank. Consequently we obtain that a linear operator on M(S) preserves zero-term rank if and only if it preserves two consecutive zero-term ranks k and k + 1, where 0 ≤ k ≤ min{m, n} − 1 if and only if it strongly preserves zero-term rank h, where 1 ≤ h ≤ min{m, n}.
Electronic Journal of Linear Algebra
Let $\S$ denote the set of symmetric matrices over some semiring, $\s$. A line of $A\in\S$ is a row or a column of $A$. A star of $A$ is the submatrix of $A$ consisting of a row and the corresponding column of $A$. The term rank of $A$ is the minimum number of lines that contain all the nonzero entries of $A$. The star cover number is the minimum number of stars that contain all the nonzero entries of $A$. This paper investigates linear operators that preserve sets of symmetric matrices of specified term rank and sets of symmetric matrices of specific star cover numbers. Several equivalences to the condition that $T$ preserves the term rank of any matrix are given along with characterizations of a couple of types of linear operators that preserve certain sets of matrices defined by the star cover number that do not preserve all term ranks.
Czechoslovak Mathematical Journal, 2000
Journal of Mathematical Sciences, 2006
We characterize linear operators on matrices over semirings that preserve the extremal cases in the bounds on term and zero-term ranks of sums and products of matrices.
Mathematics
There are many characterizations of linear operators from various matrix spaces into themselves which preserve term rank. In this research, we characterize the linear maps which preserve any two term ranks between different matrix spaces over anti-negative semirings, which extends the previous results on characterizations of linear operators from some matrix spaces into themselves. That is, a linear map T from p × q matrix spaces into m × n matrix spaces preserves any two term ranks if and only if T preserves all term ranks if and only if T is a ( P , Q , B )-block map.
Linear Algebra and its Applications, 2006
We characterize linear preservers for sets of matrix ordered tuples which satisfy extremal properties with respect to row and column ranks.
Linear and Multilinear Algebra, 2001
Zero-term rank of a matrix is the minimum number of lines (rows or columns) needed to cover all the zero entries of the given matrix. We characterize the linear operators that preserve zero-term rank of the m × n real matrices. We also obtain combinatorial equivalent condition for the zero-term rank of a real matrix.
Journal of the Korean Mathematical Society, 2008
The spanning column rank of an m×n matrix A over a semiring is the minimal number of columns that span all columns of A. We characterize linear operators that preserve the sets of matrix ordered pairs which satisfy multiplicative properties with respect to spanning column rank of matrices over semirings.
Journal of the Korean Mathematical Society
The term rank of a matrix A over a semiring is the least number of lines (rows or columns) needed to include all the nonzero entries in A. In this paper, we characterize linear operators that preserve the sets of matrix ordered pairs which satisfy extremal properties with respect to term rank inequalities of matrices over nonbinary Boolean semirings.
Linear Algebra and Its Applications, 2011
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.
Linear and Multilinear Algebra, 2017
A rank one matrix can be factored as uv t for vectors u and v of appropriate orders. The perimeter of this rank one matrix is the number of nonzero entries in u plus the number of nonzero entries in v. A matrix of rank k is the sum of k rank one matrices, a rank one decomposition. The perimeter of a matrix A of rank k is the minimum over all rank one of A of the sums of perimeters of the rank one matrices. The arctic rank of a matrix is one half the perimeter. In this article, we characterize the linear operators that preserve the symmetric arctic ranks of nonnegative symmetric matrices.
Linear Algebra and its Applications, 1987
Characterizations are obtained of those linear operators on the m x n matrices over an arbitrary semiring that preserve term rank. We also present characterizations of permanent and rook-polynomial preserving operators on matrices over certain types of semirings. Our results apply to many combinatorially interesting algebraic systems, including nonnegative integer matrices, matrices over Boolean algebras, and fuzzy matrices.
Linear Algebra and its Applications, 1997
This paper concerns a certain column rank of matrices over the nonnegative reals; we call it the spanning column rank. We have a characterization of spanning column rank 1 matrices. We also investigate the linear operators which preserve the spanning column ranks of matrices over the nonnegative part of a certain unique factorization domain in the reals. 0 Elsevier Science Inc.
Proceedings of the American Mathematical Society, 1998
The maximal column rank of an m by n matrix over a semiring is the maximal number of the columns of A which are linearly independent. We characterize the linear operators which preserve the maximal column ranks of nonnegative integer matrices.
Journal of the Korean Mathematical Society, 2005
Inequalities on the rank of the sum and the product of two matrices over semirings are surveyed. Preferences are given to the factor rank, row and column ranks, term rank, and zero-term rank of matrices over antinegative semirings.
International Scholarly Research Notices, 2015
We use the ϵ-determinant introduced by Ya-Jia Tan to define a family of ranks of matrices over certain semirings. We show that these ranks generalize some known rank functions over semirings such as the determinantal rank. We also show that this family of ranks satisfies the rank-sum and Sylvester inequalities. We classify all bijective linear maps which preserve these ranks.
Linear Algebra and its Applications, 1988
Characterizations are obtained of those linear operators over certain semirings that preserve (1) the r th coefficient of the rook polynomial, those that preserve (2) the term rank of matrices with term rank r, and those that preserve (3) the rth elementary symmetric permanental function. Our results apply to many algebraic systems of combinatorial interest, including the nonnegative integer matrices, Boolean matrices, and fuzzy matrices.
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