Séminaire Lotharingien de Combinatoire, B13g (1986).
[Formerly: Publ. I.R.M.A. Strasbourg, 1986, 316/S-13, p. 135-144.]

Norma Zagaglia

On Quasi-g-Circulant Matrices

Abstract. A matrix Q of order n is called k-quasi-g-circulant if it satisfies

PkQ = QPkg,

where P represents the permutation (1 2 ... n), (n,g)=1, and the exponents are mod n.

We prove that if (k,n)=h, a matrix Q is k-quasi-g-circulant if and only if it is h-quasi-g-circulant; then Q is a block g-circulant matrix of type (q,h), and we give a characterization for these matrices. Moreover, we define a perfect k-quasi-g-circulant permutation, and we prove that the set of these permutations is an imprimitive group of order phi(k)kqk, where phi(k) is the Euler function of k, that is the number of positive integers not greater than and prime to k.

The main part of the results of this paper have appeared in the article "On the solutions of a matrix equation," Boll. Un. Mat. Ital. 3-A (1989), 137-145.