Isolation Numbers of Integer Matrices and Their Preservers

Czechoslovak Mathematical Journal

Let Z + be the semiring of all nonnegative integers and A an m × n matrix over Z +. The rank of A is the smallest k such that A can be factored as an m × k matrix times a k × n matrix. The isolation number of A is the maximum number of nonzero entries in A such that no two are in any row or any column, and no two are in a 2 × 2 submatrix of all nonzero entries. We have that the isolation number of A is a lower bound of the rank of A. For A with isolation number k, we investigate the possible values of the rank of A and the Boolean rank of the support of A. So we obtain that the isolation number and the Boolean rank of the support of a given matrix are the same if and only if the isolation number is 1 or 2 only. We also determine a special type of m × n matrices whose isolation number is m. That is, those matrices are permutationally equivalent to a matrix A whose support contains a submatrix of a sum of the identity matrix and a tournament matrix.

Czechoslovak Mathematical Journal, 2014

Let A be a Boolean {0, 1} matrix. The isolation number of A is the maximum number of ones in A such that no two are in any row or any column (that is they are independent), and no two are in a 2 × 2 submatrix of all ones. The isolation number of A is a lower bound on the Boolean rank of A. A linear operator on the set of m × n Boolean matrices is a mapping which is additive and maps the zero matrix, O, to itself. A mapping strongly preserves a set, S, if it maps the set S into the set S and the complement of the set S into the complement of the set S. We investigate linear operators that preserve the isolation number of Boolean matrices. Specifically, we show that T is a Boolean linear operator that strongly preserves isolation number k for any 1 k min{m, n} if and only if there are fixed permutation matrices P and Q such that for X ∈ Mm,n(B) T (X) = P XQ or, m = n and T (X) = P X t Q where X t is the transpose of X.

2014

Let A be a Boolean {0, 1} matrix. The isolation number of A is the maximum number of ones in A such that no two are in any row or any column (that is they are independent), and no two are in a 2 × 2 submatrix of all ones. The isolation number of A is a lower bound on the Boolean rank of A. A linear operator on the set of m× n Boolean matrices is a mapping which is additive and maps the zero matrix, O, to itself. A mapping strongly preserves a set, S, if it maps the set S into the set S and the complement of the set S into the complement of the set S. We investigate linear operators that preserve the isolation number of Boolean matrices. Specifically, we show that T is a Boolean linear operator that strongly preserves isolation number k for any 1 6 k 6 min{m,n} if and only if there are fixed permutation matrices P and Q such that for X ∈ Mm,n(B) T (X) = PXQ or, m = n and T (X) = PXQ where X is the transpose of X.

Journal of Combinatorics, 2011

We study the functions that count matrices of given rank over a finite field with specified positions equal to zero. We show that these matrices are q-analogues of permutations with certain restricted values. We obtain a simple closed formula for the number of invertible matrices with zero diagonal, a q-analogue of derangements, and a curious relationship between invertible skew-symmetric matrices and invertible symmetric matrices with zero diagonal. In addition, we provide recursions to enumerate matrices and symmetric matrices with zero diagonal by rank, and we frame some of our results in the context of Lie theory. Finally, we provide a brief exposition of polynomiality results for enumeration questions related to those mentioned, and give several open questions.

Mathematics

Let S be an antinegative semiring. The rank of an m × n matrix B over S is the minimal integer r such that B is a product of an m × r matrix and an r × n matrix. The isolation number of B is the maximal number of nonzero entries in the matrix such that no two entries are in the same column, in the same row, and in a submatrix of B of the form b i , j b i , l b k , j b k , l with nonzero entries. We know that the isolation number of B is not greater than the rank of it. Thus, we investigate the upper bound of the rank of B and the rank of its support for the given matrix B with isolation number h over antinegative semirings.

2006

We characterize the linear operators which preserve the factor rank of integer matrices. That is, if M is the set of all m £ n matrices with entries in the integers and min(m;n) > 1, then a linear operator T on M preserves the factor rank of all matrices in M if and only if T has the form either T(X) = UXV for all X 2 M, or m = n and T(X) = UX t V for all X 2 M, where U and V are suitable nonsingular integer matrices. Other characterizations of factor rank-preservers of integer matrices are also given.

Communications of the Korean Mathematical Society, 2007

The maximal column rank of an m × n matrix A over the ring of integers, is the maximal number of the columns of A that are weakly independent. We characterize the linear operators that preserve the maximal column ranks of integer matrices.

Communications of the Korean Mathematical Society, 2005

The set of all m × n matrices with entries in Z + is denoted by Mm×n(Z+). We say that a linear operator T on Mm×n(Z+) is a (U, V)-operator if there exist invertible matrices U ∈ M m×m (Z +) and V ∈ M n×n (Z +) such that either T (X) = U XV for all X in Mm×n(Z+), or m = n and T (X) = U X t V for all X in Mm×n(Z+). In this paper we show that a linear operator T preserves the rank of matrices over the nonnegative integers if and only if T is a (U, V)operator. We also obtain other characterizations of the linear operator that preserves rank of matrices over the nonnegative integers.

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.

Kyungpook mathematical journal, 2013

In this paper, we consider the row rank inequalities derived from comparisons of the row ranks of the additions and multiplications of nonnegative integer matrices and construct the sets of nonnegative integer matrix pairs which is occurred at the extreme cases for the row rank inequalities. We characterize the linear operators that preserve these extreme sets of nonnegative integer matrix pairs * Corresponding Author.