RANK-PRESERVING OPERATORS OF NONNEGATIVE INTEGER MATRICES

Title & Authors
RANK-PRESERVING OPERATORS OF NONNEGATIVE INTEGER MATRICES
SONG, SEOK-ZUN; KANG, KYUNG-TAE; JUN, YOUNG-BAE;

Abstract
The set of all $\small{m\;{\times}\;n}$ matrices with entries in $\small{\mathbb{Z}_+}$ is de­noted by $\small{\mathbb{M}{m{\times}n}(\mathbb{Z}_+)}$. We say that a linear operator T on $\small{\mathbb{M}{m{\times}n}(\mathbb{Z}_+)}$ is a (U, V)-operator if there exist invertible matrices $\small{U\;{\in}\; \mathbb{M}{m{\times}n}(\mathbb{Z}_+)}$ and $\small{V\;{\in}\;\mathbb{M}{m{\times}n}(\mathbb{Z}_+)}$ such that either T(X) = UXV for all X in $\small{\mathbb{M}{m{\times}n}(\mathbb{Z}_+)}$, or m = n and T(X) = $\small{UX^{t}V}$ for all X in $\small{\mathbb{M}{m{\times}n}(\mathbb{Z}_+)}$. 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.
Keywords
semidomain;(U, V)-operator;rank preserver;
Language
English
Cited by
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