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linear operator; rank; perimeter; $(P, Q, B)$-operator
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.
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