On a family of tridiagonal matrices

TL;DR

This paper explores a unique correspondence between positive definite symmetric tridiagonal matrices of determinant n and elements of the multiplicative group modulo n, and also discusses a construction linking matrices to sequences of integral polytopes.

Contribution

It establishes a novel one-to-one correspondence between certain matrices and algebraic elements, and introduces a new method to associate polytopes with matrices.

Findings

Matrices of determinant n correspond to elements of (/n)^*

Properties of this correspondence are analyzed

A construction linking matrices to polytopes is proposed

Abstract

We show that certain integral positive definite symmetric tridiagonal matrices of determinant $n$ are in one to one correspondence with elements of $(\mathbb Z/n\mathbb Z)^*$. We study some properties of this correspondence. In a somewhat unrelated second part we discuss a construction which associates a sequence of integral polytopes to every integral symmetric matrix.