Balanced matrices

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.