Spectral symmetry in conference matrices

Willem H. Haemers*, Leila Parsaei Majd

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review


A conference matrix of order n is an n× n matrix C with diagonal entries 0 and off-diagonal entries ± 1 satisfying CC= (n- 1) I. If C is symmetric, then C has a symmetric spectrum Σ (that is, Σ = - Σ) and eigenvalues ±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.
Original languageEnglish
JournalDesigns, Codes, and Cryptography
Publication statusE-pub ahead of print - Mar 2021


  • Conference matrix
  • Paley graph
  • Seidel matrix
  • Signed graph
  • Symmetric spectrum


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