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Balanced matrices

Theophilus Agama, Gael Kibiti

TL;DR

It turns out that there does exist a direct communication among the leading entry, the trace, determinants and, hence, the eigen-values of these matrices of order $2\times 2$.

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.

Balanced matrices

TL;DR

It turns out that there does exist a direct communication among the leading entry, the trace, determinants and, hence, the eigen-values of these matrices of order .

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 . These matrices have an interesting property that allows us to predict their quadratic forms using their spectrum, without an information about their entries.

Paper Structure

This paper contains 8 sections, 8 theorems, 54 equations.

Table of Contents

  1. Introduction and motivation
  2. Motivation
  3. Balanced matrices
  4. Elementary properties of fully balanced matrices
  5. Trace, determinants and eigenvalues associated with balanced matrices
  6. Discrepancies of fully-balanced matrices
  7. Quadratic forms associated with balanced matrices
  8. Further remarks

Key Result

Theorem 4.1

Let $A, B\in \mathbb{M}_{n}(\mathbb{R})$ be fully-balanced matrices and let $\lambda\in \mathbb{R}$. The following hold:

Theorems & Definitions (34)

  • Definition 3.1
  • Example 3.2
  • Theorem 4.1
  • proof
  • Proposition 5.1
  • proof
  • Remark 5.2
  • Remark 5.3
  • Theorem 5.4
  • proof
  • ...and 24 more