On orthogonal circulant matrices

A square matrix is called circulant if each of its rows is obtained as a cyclic permutation of the previous row by one position to the right. In our talk, we are concerned with orthogonal circulant matrices of order N with off-diagonal terms taken from the set {-1,1}. We derive necessary conditions on the diagonal terms in those matrices, demonstrating in particular that their value cannot exceed N/2-1. We prove that any orthogonal matrix of order N having N/2-1 on the diagonal and elements from {-1,1} off the diagonal is either similar to a circulant matrix with the same property, or equal to A+dI, where A is antisymmetric. Finally, we find an explicit characterization of all circulant orthogonal matrices with off-diagonal terms taken from {-1,1} up to the order N=50.

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