Isolation numbers of integer matrices and their preservers
Bull. Korean Math. Soc. 2020 Vol. 57, No. 3, 535-545
Published online February 14, 2020
Printed May 31, 2020
LeRoy B. Beasley, Kyung-Tae Kang, Seok-Zun Song
Utah State University; Jeju National University; Jeju National University
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 $A$.
Keywords : Isolation number, upper ideal, linear operator, $(P,Q)$-operator
MSC numbers : 15A86, 15A04, 15A33
Supported by : This research was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education(No.2016R1D1A1B02006812).
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