Inseparable orthogonal matrices over \(\mathbb{Z}_2\)
type: Item
instance of: scholarly article
title: Inseparable orthogonal matrices over \(\mathbb{Z}_2\)
zbMATH Open document ID: 0921.15008
author: Guo-Hui Zhang
published in: Journal of Combinatorial Theory. Series A
publication date: 1999-03-30T00:00:00Z
review text: A \((0,1)\)-matrix \(A\) is called orthogonal over \(\mathbb{Z}_2\) if both \(AA^T\) and \(A^TA\) are diagonal matrices. A marix \(A\) is called inseparable if \(A\) contains no zero row or zero column and there do not exist permutation matrices \(P\) and \(Q\) such that \[ PAQ= \begin{pmatrix} B&0\\ 0&C\end{pmatrix}. \] A matrix \(A\) is said to be of type 0 if \(AA^T= 0\) and \(A^TA=0\). A square matrix \(A\) of order \(n\) is said to be of type 1 if \(AA^T=I_n\). It turns out that an inseparable orthogonal matrix over \(\mathbb{Z}_2\) is either of type 0 or of type 1. Let \(f_0(m,n)\) (respectively, \(F_0(m,n)\)) denote the smallest (respectively, largest) number of 1's in an \(m\times n\) inseparable orthogonal matrix of type 0 over \(\mathbb{Z}_2\), and \(f_1(n)\) (respectively, \(F_1(n)\)) denote the smallest (respectively largest) number of 1's in an \(n\times n\) inseparable orthogonal matrix of type 1 over \(\mathbb{Z}_2\). The formulas for \(f_0(m,n)\), \(F_0(m,n)\), \(f_1(n)\), and \(F_1(n)\) are completely determined.
Mathematics Subject Classification ID: 15B36
Mathematics Subject Classification ID: 05B20
zbMATH DE Number: 1236146
zbMATH Keywords: \((0,1)\)-matrix
zbMATH Keywords: inseparable orthogonal matrix
reviewed by: Rodica Covaci
MaRDI profile type: Publication
full work available at URL: jcta.1997.2854
OpenAlex ID: W2074220840
cites work: Q4281318
cites work: Q4198777
cites work: Combinatorially orthogonal matrices and related graphs
cites work: Edge sets contained in circuits
DOI: 10.1006/JCTA.1997.2854
description: scientific article; zbMATH DE number 1236146