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 $< u,v >=0$ 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 $n$.

💡 Research Summary

The paper addresses the spectral analysis of a family of matrices defined over the ring of integers modulo m, denoted ℤₘ. For integers m, n > 1, the authors consider the projective (n‑1)-space ℙₙ,ₘ, which consists of the non‑zero vectors in ℤₘⁿ modulo scalar multiplication by units of ℤₘ. They define a binary matrix Aₙ,ₘ whose rows and columns are indexed by the points of ℙₙ,ₘ, with entry a_{uv}=1 if the modular inner product ⟨u,v⟩ equals zero, and a_{uv}=0 otherwise. The central object of study is the symmetric matrix Bₙ,ₘ = Aₙ,ₘ Aₙ,ₘᵗ, which records how many common orthogonal neighbours each pair of points shares.

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