On the eigenvalues of certain matrices over Zm

Let m,n>1 be integers, and let Pn,m be the point set of the projective (n−1)-space (defined by Chee and Ling (1993) [2]) over the ring Zm of integers modulo m. Let An,m=(auv) be the matrix with rows and columns being labeled by elements of Pn,m, where auv=1 if the inner product 〈u,v〉=0 and auv=0 otherwise. Let Bn,m=An,mAn,mt. The eigenvalues of Bn,m have been studied by Alon (1986) [1], Chee and Ling [2] and Chee et al. [3], where their applications in the study of expanders and locally decodable codes were described. In this paper, we completely determine the eigenvalues of Bn,m for general integers m and n.