On the eigenvalues of certain matrices over

Let m , n > 1 be integers, and let P n , m be the point set of the projective ( n − 1 ) -space (defined by Chee and Ling (1993) [2]) over the ring Z m of integers modulo m . Let A n , m = ( a u v ) be the matrix with rows and columns being labeled by elements of P n , m , where a u v = 1 if the inner product 〈 u , v 〉 = 0 and a u v = 0 otherwise. Let B n , m = A n , m A n , m t . The eigenvalues of B n , 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 B n , m for general integers m and n .