On orthogonal matrices with zero diagonal

TL;DR

This paper characterizes the existence of real orthogonal matrices with zero diagonals, providing constructions from tournaments and applying results to graph eigenvalues, revealing new structural insights.

Contribution

It establishes existence conditions for orthogonal matrices with zero diagonals, including symmetric cases, and connects these to graph eigenvalue problems.

Findings

Existence of OMZD(n) for all n except 1 and 3.

Symmetric OMZD(n) exists iff n is even and not 4.

Construction from doubly regular tournaments.

Abstract

We consider real orthogonal $n\times n$ matrices whose diagonal entries are zero and off-diagonal entries nonzero, which we refer to as $\mathrm{OMZD}(n)$. We show that there exists an $\mathrm{OMZD}(n)$ if and only if $n\neq 1,\ 3$, and that a symmetric $\mathrm{OMZD}(n)$ exists if and only if $n$ is even and $n\neq 4$. We also give a construction of $\mathrm{OMZD}(n)$ obtained from doubly regular tournaments. Finally, we apply our results to determine the minimum number of distinct eigenvalues of matrices associated with some families of graphs, and consider the related notion of orthogonal matrices with partially-zero diagonal.