A Matrix Analogue of Schur-Siegel-Smyth Trace Problem

TL;DR

This paper extends the classical Schur-Siegel-Smyth trace problem to matrices, establishing lower bounds for traces of eigenvalues of certain positive-definite matrices and identifying their limit points.

Contribution

It introduces a matrix analogue of the trace problem, providing optimal bounds and limit points for traces of eigenvalues of symmetrizable integer matrices.

Findings

Established the best possible lower bound for Tr_2(A) as 6n - 5.

Proved the smallest limit point of the normalized trace Tr_2(A)/n is 6.

Extended Smyth's density results to symmetric positive definite matrices.

Abstract

Let $\mathcal{S}$ be the set of all positive-definite, symmetrizable integer matrices with non-zero upper and lower diagonal and $\mathcal{T}$ to be the set of all positive-definite real symmetric matrices with nonzero upper diagonal such that all non-zero entries are square-roots of some positive integers and the matrices satisfy a certain cycle condition. In this paper, for any $n \times n$ matrix $A \in \mathcal{S} \cup \mathcal{T}$ and any $k \in \mathbb{N}$ we find a general lower bound for $Tr_{2^k}(A)$, i.e, the sum of $2^k$-th power of eigenvalues of $A$, which depends on $n$ as well as some other variables. In particular, we obtain the best possible lower bound for $Tr_2(A) $ that is $6n - 5$. As a strong outcome of this result we show that the smallest limit point of $\overline{Tr_2(A)} = \frac{Tr_2(A)}{n}$ is $6$. This is a solution of an analogue of ``Schur - Siegel - Smyth trace problem" for characteristic polynomials of matrices in $\mathcal{S} \cup \mathcal{T}$. We also obtain a lower bound of smallest limit point of $\overline{Tr_{2^k}(A)}$ for any positive integer $k > 1$ and for the same set of matrices. Furthermore, we exhibit that the famous results of Smyth on density of absolute trace measure and absolute trace-2 measure of totally positive integers are also true for the set of symmetric integer connected positive definite matrices.