Linear Operators That Preserve Arctic Ranks of Boolean Matrices

Kyung-Tae Kang,Seok-Zun Song

doi:10.3390/math9101087

Abstract

We study some properties of arctic rank of Boolean matrices. We compare the arctic rank with Boolean rank and term rank of a given Boolean matrix. Furthermore, we obtain some characterizations of linear operators that preserve arctic rank on Boolean matrix space.

Highlights

Introduction and PreliminariesCitation: Kang, K.-T; Song, S.-Z.Linear Operators That Preserve ArcticRanks of Boolean Matrices
We show that the arctic rank of a matrix is equal or greater than both the Boolean rank (Theorem 1) and the term rank (Theorem 2)
We study the basic behavior of the arctic rank of Boolean matrices

Summary

Introduction

Introduction and PreliminariesLinear Operators That Preserve ArcticRanks of Boolean Matrices. If A0 is a Boolean matrix which is obtained from A by deleting some zero rows or columns, ar( A0 ) = ar( A); ar( At ) = ar( A); if P ∈ Mm and Q ∈ Mn are permutation matrices, ar( PAQ) = ar( A); if A is a sum of h(≥ 2) cells which are collinear, ar( A) = 1+2 h . For k = 1, 32 , 2, 52 , 3, · · · , let A R(k) denote the set of Boolean matrices in Mm,n whose arctic rank is k.

Results

Conclusion