Spectral symmetry in conference matrices

A conference matrix of order \(n\) is an \(n\times n\) matrix \(C\) with diagonal entries \(0\) and off-diagonal entries \(\pm 1\) satisfying \(CC^\top=(n-1)I\). If \(C\) is symmetric, then \(C\) has a symmetric spectrum \(\Sigma\) (that is, \(\Sigma=-\Sigma\)) and eigenvalues \(\pm\sqrt{n-1}\). We...

Full description

Saved in:
Bibliographic Details
Published in:arXiv.org 2021-01
Main Authors: Haemers, Willem H, Leila Parsaei Majd
Format: Article
Language:English
Subjects:
Eigenvalues
Graphs
Mathematical analysis
Matrix methods
Symmetry
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:A conference matrix of order \(n\) is an \(n\times n\) matrix \(C\) with diagonal entries \(0\) and off-diagonal entries \(\pm 1\) satisfying \(CC^\top=(n-1)I\). If \(C\) is symmetric, then \(C\) has a symmetric spectrum \(\Sigma\) (that is, \(\Sigma=-\Sigma\)) and eigenvalues \(\pm\sqrt{n-1}\). We show that many principal submatrices of \(C\) also have symmetric spectrum, which leads to examples of Seidel matrices of graphs (or, equivalently, adjacency matrices of complete signed graphs) with a symmetric spectrum. In addition, we show that some Seidel matrices with symmetric spectrum can be characterized by this construction.
ISSN:2331-8422