On the Eigenvalues of Certain Matrices Over $\mathbb{Z}_m$

Let m,n > 1 be integers and Pn,m be the point set of the projective (n − 1)-space (defined by [2]) over the ring Zmof integers modulo m. Let An,m = (auv) be the matrix with rows and columns being labeled by Pn,m, where auv = 1 if the inner product 〈u, v〉 = 0 and auv = 0 otherwise. Let Bn,m = An,mA t n,m. The eigenvalues of Bn,m have been studied by [1, 2, 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.