Incidence matrices of finite projective planes and their eigenvalues
type: Item
instance of: scholarly article
title: Incidence matrices of finite projective planes and their eigenvalues
zbMATH Open document ID: 0885.05042
author: John G. Thompson
published in: Journal of Algebra
publication date: 1998-02-18T00:00:00Z
review text: The author provides two results concerning the incidence matrix of a finite projective plane. Let \(A\) be the 0-1 incidence matrix of a projective plane of order \(n\). \(A\) will be viewed as a square matrix of order \(N\) (\(=n^2+n+1\)) over \(\mathbb{C}\), the complex numbers. Set \(f_A(x)=\text{det}(xI-A)\), and let \(F_A\) be the splitting field of \(f_A(x)\) with \(F_A\subseteq\mathbb{C}\), and set \(G_A=\text{Gal}(F_A/\mathbb{Q})\). Let \(c_A\) be that element of \(G_A\) determined by complex conjugation, so that \(c_A(\alpha)=\overline\alpha\) for every zero \(\alpha\) of \(f_A(x)\) in \(\mathbb{C}\). We say that \(A\) splits if and only if \(c_A\) is of order 2, and \(\langle c_A\rangle\) is a direct factor of \(G_A\). Let \({\mathcal T}_N\) denote the set of all permutation matrices of \(\text{GL} (N,\mathbb{C})\) which represent transpositions. The first result is: Theorem 1. If \(n\) is not a square and \(A\) is not split, then \(AT\) is split for all \(T\in{\mathcal T}_N.\) For any incidence matrix \(A\), write \(f_A(x)=(x-(n+1))g_A(x)\). Then \(g_A(x)\) is a monic polynomial with integer coefficients such that \(g_A(n+1)\neq 0\). The second result is the following interlacing theorem: Theorem 2. Suppose \(n\) is not a square and \(T\in{\mathcal T}_N\). Let \(g_A(x)=D(x)E(x)\) and \(g_{AT}(x)=D(x)F(x)\), where \(D(x)\) is the greatest common divisor of \(g_A(x)\) and \(g_{AT}(x)\). Then \(E(x)\) and \(F(x)\) are square free, and on the circle of radius \(\sqrt n\) centered at the origin, between any two successive zeroes of \(E(x)\) there is precisely one zero of \(F(x)\). The author remarks that with some mild modifications, the above results can be adapted to the case where \(n\) is a square.
Mathematics Subject Classification ID: 05B25
Mathematics Subject Classification ID: 05B20
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 51E15
zbMATH DE Number: 1019696
zbMATH Keywords: eigenvalues
zbMATH Keywords: incidence matrix
zbMATH Keywords: interlacing
zbMATH Keywords: projective planes
reviewed by: William E. Cherowitzo
MaRDI profile type: Publication
full work available at URL: jabr.1996.6919
OpenAlex ID: W2032920668
DOI: 10.1006/JABR.1996.6919
description: scientific article; zbMATH DE number 1019696