# Possible Isolation Number of a Matrix Over Nonnegative Integers

Article

## Journal/Book Title/Conference

Czechoslovak Mathematical Journal

## Publisher

Springer Berlin Heidelberg

5-8-2018

1

12

## Abstract

Let ℤ+ be the semiring of all nonnegative integers and A an m × n matrix over ℤ+. The rank of A is the smallest k such that A can be factored as an m × k matrix times a k×n matrix. The isolation number of A is the maximum number of nonzero entries in A such that no two are in any row or any column, and no two are in a 2 × 2 submatrix of all nonzero entries. We have that the isolation number of A is a lower bound of the rank of A. For A with isolation number k, we investigate the possible values of the rank of A and the Boolean rank of the support of A. So we obtain that the isolation number and the Boolean rank of the support of a given matrix are the same if and only if the isolation number is 1 or 2 only. We also determine a special type of m×n matrices whose isolation number is m. That is, those matrices are permutationally equivalent to a matrix A whose support contains a submatrix of a sum of the identity matrix and a tournament matrix.

The original publication is available at: https://articles.math.cas.cz/10.21136/CMJ.2018.0068-17

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#### DOI

https://doi.org/10.21136/CMJ.2018.0068-17