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

Let $m,n>1$ be integers and $\mathbb{P}_{n,m}$ be the point set of the projective $(n-1)$-space (defined by [2]) over the ring $\mathbb{Z}_m$of integers modulo $m$. Let $A_{n,m}=(a_{uv})$ be the matrix with rows and columns being labeled by elements of $\mathbb{P}_{n,m}$, where $a_{uv}=1$ if the inner product $ =0$ and $a_{uv}=0$ otherwise. Let $B_{n,m}=A_{n,m}A_{n,m}^t$. The eigenvalues of $B_{n,m}$ 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 $B_{n,m}$ for general integers $m$ and $n$.