On the ranks of skew centrosymmetric matrices over finite fields
type: Item
instance of: scholarly article
title: On the ranks of skew centrosymmetric matrices over finite fields
zbMATH Open document ID: 0909.15006
author: William C. Waterhouse
published in: Finite Fields and their Applications
publication date: 1999-03-28T00:00:00Z
review text: Let \(F\) be a field of characteristic different of 2. A skew centrosymmetric \(n\times n\) matrix \(A\) is a skew matrix which is also symmetric around the secondary diagonal, i.e., satisfying \(a_{ij}=a_{ji}\) and \(a_{ij}=a_{n+1-j,n+1-i}\). In this paper it is shown that any skew centrosymmetric matrix \(A\) is similar to a matrix of form \(\left(\begin{smallmatrix} 0&C\\ D&0\end{smallmatrix}\right)\). If \(n=2k\), then \(C\) is \(k\times k\) and \(D=-C^T\). If \(n=2k+1\), then \(C\) is \(k\times(k+1)\) and \(D=-C^T\cdot\text{diag}(1,\dots,1,1/2)\). Hence the rank of \(A\) is equal to twice the rank of \(C\). This result yields a formula for the number of skew centrosymmetric matrices of any particular rank over a finite field \(F\).
Mathematics Subject Classification ID: 15B33
Mathematics Subject Classification ID: 15A03
zbMATH DE Number: 1206737
zbMATH Keywords: skew centrosymmetric matrix
zbMATH Keywords: finite field
zbMATH Keywords: rank of a matrix
MaRDI profile type: Publication
full work available at URL: ffta.1997.0196
OpenAlex ID: W2050200623
cites work: On the ranks of skew-centrosymmetric matrices over finite fields
DOI: 10.1006/FFTA.1997.0196
description: scientific article; zbMATH DE number 1206737