Highly symmetric generalized circulant permutation matrices

In this paper we investigate generalized circulant permutation matrices of composite order. We give a complete characterization of the order and the structure of symmetric generalized k-circulant permutation matrices in terms of circulant and retrocirculant block (0, 1)-matrices in which each block contains exactly one or two entries 1. In particular, we prove that a generalized k-circulant matrix A of composite order n = km is symmetric if and only if either k = m − 1 or k ≡ 0 or k ≡ 1 mod m, and we obtain three basic symmetric generalized k-circulant permutation matrices, from which all others are obtained via permutations of the blocks or by direct sums. Furthermore, we extend the characterization of these matrices to centrosymmetric matrices.