Abstract:We investigate the perimeter of nonnegative integer matrices. We also characterize the linear operators which preserve the rank and perimeter of nonnegative integer matrices. That is, a linear operator $T$ preserves the rank and perimeter of rank-$1$ matrices if and only if it has the form $T(A)=P(A\circ B)Q$, or $T(A)=P(A^t \circ B)Q $ with appropriate permutation matrices $P$ and $Q$ and positive integer matrix $B$, where $\circ $ denotes Hadamard product.
Keywords: linear operator, rank, perimeter, $(P,Q,B)$-operator
AMS Subject Classification: 15A04, 15A33, 15A48