On the eigenvalues of certain matrices over \(\mathbb Z_m\) (Q1942052)

Let \(m, n > 1\) be integers. Let \(\mathbb{Z}_m\) be the ring of integers modulo \(m\), let \(\mathbb{Z}_m^{*}\) be the group of units of \(\mathbb{Z}_m\) and let \(\mathbb{Z}_m^n\) be the set of \(n\)-tuples with entries in \(\mathbb{Z}_m\). It is said that \(u,v \in \mathbb{Z}_m^n\) are equivalent (denoted by \(u \sim v\)) if there exists a \(\lambda \in \mathbb{Z}_m^{*}\) such that \(u_i = \lambda v_i\) for every \(i \in \{1,2,\dots,n\}\). Let \(\mathbb{S}_{n,m}=\{ u \in \mathbb{Z}_m^n : \text{gcd}(u_1,u_2, \dots,u_n,m)=1 \}\), where gcd is the greatest common divisor, and let \(\mathbb{P}_{n,m}\) be the set of equivalence classes of elements of \(\mathbb{S}_{n,m}\) under \(\sim\). 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 \(\langle u,v \rangle =0\) and \(a_{uv}=0\) otherwise, and consider \(B_{n,m}=A_{n,m} A_{n,m}^T\). In this paper, the author completely determines the eigenvalues of matrix \(B_{n,m}\) for any positive integer \(m\), completing the results of different researchers. First, he analyzes the prime power case and after the general case.