Balanced matrices

TL;DR

This paper introduces the concept of balanced matrices, exploring their properties, determinants, eigenvalues, and how their spectrum can predict quadratic forms without entry details.

Contribution

It defines balanced matrices and analyzes their properties, especially relating determinants, eigenvalues, and spectrum, providing new insights into their structure and behavior.

Findings

Balanced matrices have a predictable relationship between spectrum and quadratic forms.

For 2x2 balanced matrices, there is a direct link between leading entry, trace, and eigenvalues.

Spectrum can be used to predict quadratic forms without knowing matrix entries.

Abstract

In this paper, we introduce a particular class of matrices. We study the concept of a matrix to be \emph{balanced}. We study some properties of this concept in the context of matrix operations. We examine the behaviour of various matrix statistics in this setting. The crux will be to understand the determinants and the eigenvalues of balanced matrices. It turns out that there exist a direct communication among the leading entry, the trace, determinants and, hence, the eigenvalues of these matrices of order $2\times 2$. These matrices have an interesting property that allows us to predict their quadratic forms using their spectrum, without an information about their entries.