Highly symmetric generalized circulant permutation matrices

In this paper we investigate generalized circulant permutation ma- trices of composite order. We give a complete characterization of the order and the structure of symmetric generalized k{circulant permu- tation 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 com- posite order n = km is symmetric if and only if either k = m 1 or k 0 or k 1 modulo m, and we obtain three basic symmetric gener- alized k{circulant permutation matrices, from which all others are ob- tained via permutations of the blocks or by direct sums. Furthermore, we extend the characterization of these matrices to centrosymmetric matrices.