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

TL;DR

This paper determines the eigenvalues of a specific matrix associated with projective spaces over integers modulo m, extending previous work and enabling applications in expanders and coding theory.

Contribution

It provides a complete characterization of the eigenvalues of matrices related to projective spaces over rac{m} and rac{n}, generalizing earlier partial results.

Findings

Eigenvalues of B_{n,m} are explicitly determined for all m and n.

Results have implications for the study of expanders and locally decodable codes.

Generalization of previous eigenvalue results to broader classes of matrices.

Abstract

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 $n$.