On Orthogonal Matrices with Zero Diagonal

This paper considers real orthogonal $n\times n$ matrices whose diagonal entries are zero and off-diagonal entries nonzero, which are referred to as $\OMZD(n)$. It is shown that there exists an $\OMZD(n)$ if and only if $n\neq 1,\ 3$, and that a symmetric $\OMZD(n)$ exists if and only if $n$ is even and $n\neq 4$. Also, a construction of $\OMZD(n)$ obtained from doubly regular tournaments is given. Finally, the results are applied to determine the minimum number of distinct eigenvalues of matrices associated with some families of graphs, and the related notion of orthogonal matrices with partially-zero diagonal is considered.