Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

Article

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Bull. Korean Math. Soc. 2020; 57(3): 535-545

Online first article February 14, 2020      Printed May 31, 2020

https://doi.org/10.4134/BKMS.b180210

Copyright © The Korean Mathematical Society.

Isolation numbers of integer matrices and their preservers

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).