Rank-Preserving Operators of Nonnegative Integer Matrices
Seok-Zun Song2005, Communications of the Korean Mathematical Society
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Abstract
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.
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