ISOLATION NUMBERS OF INTEGER MATRICES AND THEIR PRESERVERS

Let A be an m × n matrix over nonnegative integers. The isolation number of A is the maximum number of isolated entries in A. We investigate linear operators that preserve the isolation number of matrices over nonnegative integers. We obtain that T is a linear operator that strongly preserve isolation number k for 1 ≤ k ≤ min{m,n} if and only if T is a (P,Q)-operator, that is, for fixed permutation matrices P and Q, T (A) = PAQ or, m = n and T (A) = PAtQ for any m× n matrix A, where At is the transpose of A.