On Orthogonal Matrices with Zero Diagonal

Abstract

We consider real orthogonal n × n matrices whose diagonal entries are zero and offdiagonal entries nonzero, which we refer to as OMZD(n). We show that there exists an OMZD(n) if and only if n = 1, 3, and that a symmetric OMZD(n) exists if and only if n is even and n = 4. We also give a construction of 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.

Sign patterns of orthogonal matrices and the strong inner product property

“…For a bipartite graph G with bipartition α ∪ β let B(G) be the set of m × n matrices B, with rows and columns indexed by α and β respectively, such that the (i, j) entry of B is nonzero if and only if i and j ′ are adjacent in G. Let G n denote the graph obtained from the complete bipartite graph K n,n by deleting a perfect matching. Using Lemma 5.10 and Theorem 5.13 the authors of [13] show the following.…”

“…By Theorem 4.5 we have the following corollary. Recently hollow orthogonal matrices were used in [13] to study the minimum number of distinct eigenvalues of certain families of symmetric matrices. In doing so they used the following construction.…”

“…The proof is by induction on n. It is easy to verify that hollow orthogonal matrices of order n = 1, 3 do not exist and that a non-symmetric hollow orthogonal matrix of order n = 2 does not exist. Examples of hollow orthogonal matrices, not equivalent to a symmetric matrix, of order n = 4, 5 are provided in [13]. In particular, for n = 4 we have…”

“…In order to understand the role of hollow orthogonal matrices in [13] we require a few definitions. Let A ∈ Sym n .…”

“…Let G n,k denote the bipartite graph obtained by deleting a matching of size k from K n,n . In order to establish that q(G n,k ) = 2 the authors of [13] constructed many orthogonal matrices. However, this observation is a simple consequence of the SIPP.…”