On orthogonal matrices with zero diagonal

2018/10/21 by Bailey, Robert F., Craigen, Robert · 3 citations
#05B20 #05C50 #15A18 #15B10 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1810.08961

Abstract

We consider real orthogonal n× n matrices whose diagonal entries are zero and off-diagonal 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.

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