Perimeter preservers of nonnegative 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.

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.

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.

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, 2007

Let m, n and k be positive integers such that 2 k < n m. Let V denote either the vector space of all m × n matrices over a field with at least three elements or the vector space of all n × n Hermitian matrices over a field F of characteristic / = 2 associated with an involution. We characterize surjective mappings T from V onto itself such that for every pair A, B ∈ V , rank(A -B) k if and only if rank(T (A) -T (B)) k.

Czechoslovak Mathematical Journal, 2000

Linear and Multilinear Algebra, 2001

We characterize those linear operators on triangular or diagonal matrices preserving the numerical range or radius.

Rocky Mountain Journal of Mathematics, 2006

Let A and B be n × n matrices. A classical result about the rank function is Sylvester's inequality which states that the rank of the product of AB is at most min{rank (A), rank (B)} and at least rank (A) + rank (B) − n. A generalization of Sylvester's inequality is Frobenius's inequality which states that rank (AB) + rank (BC) ≤ rank (ABC) + rank (B).

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.

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.