An upper bound for the determinant of a diagonally balanced symmetric matrix

TL;DR

This paper proves a conjectured inequality that bounds the determinant of a diagonally balanced symmetric matrix relative to the product of its diagonal entries, providing a new theoretical limit for such matrices.

Contribution

It establishes a proven upper bound for the determinant of diagonally balanced symmetric matrices, confirming a previously conjectured inequality.

Findings

Proved the determinantal inequality for diagonally balanced symmetric matrices.

Established the explicit upper bound involving matrix size n.

Confirmed the conjecture for all n ≥ 2.

Abstract

We prove a conjectured determinantal inequality: \frac{\det J}{\prod_{i=1}^nJ_{ii}}\le 2(1-\frac{1}{n-1})^{n-1}, where $J$ is a real $n\times n$ ($n\ge 2$) diagonally balanced symmetric matrix.