# Topological Properties of J-Orthogonal Matrices

12-16-2020

Dissertation

## Degree Name

Doctor of Philosophy (PhD)

## Department

Mathematics and Statistics

Frank J. Hall

## Abstract

Let Mn be the set of all n×n real matrices. A matrix J ∈ Mn is said to be a signature matrix if J is diagonal and its diagonal entries are±1. If J is a signature matrix, a nonsingular matrix A ∈ Mn is said to be a J-orthogonal matrix if A>JA = J. Let Ωn be the set of all n×n, J-orthogonal matrices. Part 2 of this dissertation includes a straightforward proof of the known topological result that for J 6= ±I, the set of all n×n J-orthogonal matrices has four connected components. An important tool in this analysis is Proposition 3.2.1 on the characterization of J-orthogonal matrices in the paper “J-orthogonal matrices: properties and generation”, SIAM Review 45 (3) (2003), 504–519, by Higham. The expression of the four components allows formulation of some further noteworthy properties. For example, it is shown that the four components are homeomorphic and group isomorphic, and that each component has exactly 2n−2 signature matrices. In Part 3 of this dissertation, the standard linear operators T : Mn → Mn that strongly preserve J-orthogonal matrices, i.e. T(A) is J-orthogonal if and only if A is J-orthogonal are characterized. The material in Part 2 of this dissertation is contained in the article “Topological properties of J-orthogonal matrices”, Linear and Multilinear Algebra, 66 (2018), 2524-2533, by S. Motlaghian, A. Armandnrjad, F. J. Hall. The material in Part 3 of this dissertation is contained in the article “Topological properties of J-orthogonal matrices, Part II”, Linear and Multilinear Algebra, doi: 10.1080/03081087.2019.1601667 by S. Motlaghian, A. Armandnrjad, F. J. Hall. In Part 4 of this dissertation some connections between J-orthogonal and G-matrices are investigated.

## DOI

https://doi.org/10.57709/20410419