Isolation numbers of integer matrices and their preservers
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Let $A$ be an $m\times n$ 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 $1\le k\le \min\{m,n\}$ 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 $T(A) = PA^t Q$ for any $m\times n$ matrix $A$, where $A^t $ is the transpose of $A$.
Abstract
Let $A$ be an $m\times n$ 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 $1\le k\le \min\{m,n\}$ 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 $T(A) = PA^t Q$ for any $m\times n$ matrix $A$, where $A^t $ is the transpose of $A$.
참고문헌 (14)
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