Quasi-orthogonal extension of symmetric matrices

TL;DR

This paper explores the concept of quasi-orthogonal extensions for symmetric matrices, expanding previous research from skew-symmetric matrices using a novel approach to understand their properties and potential applications.

Contribution

It introduces a new approach to study quasi-orthogonal extensions specifically for symmetric matrices, broadening the scope of prior work on skew-symmetric matrices.

Findings

Characterization of quasi-orthogonal extensions for symmetric matrices

Comparison with previous results on skew-symmetric matrices

Potential applications in matrix theory and related fields

Abstract

An $n\times n$ real matrix $Q$ is quasi-orthogonal if $Q^{\top}Q=qI_{n}$ for some positive real number $q$. If $M$ is a principal sub-matrix of a quasi-orthogonal matrix $Q$, we say that $Q$ is a quasi-orthogonal extension of $M$. In a recent work, the authors have investigated this notion for the class of real skew-symmetric matrices. Using a different approach, this paper addresses the case of symmetric matrices.