Self-adjoint Elements in the Pseudo-unitary Group ${\bf U}\left(p,p\right)$

TL;DR

This paper characterizes the set of Hermitian, self-adjoint elements within the pseudo-unitary group ${f U}(p,p)$, which preserve an indefinite metric on complex vector spaces, providing insights into their structure.

Contribution

It provides a detailed description of the self-adjoint elements in ${f U}(p,p)$, a topic not extensively explored before.

Findings

Characterization of Hermitian elements in ${f U}(p,p)$

Structural insights into ${f U}_{s}(p,p)$

Mathematical description of self-adjoint pseudo-unitary matrices

Abstract

The pseudo-unitary group ${\bf U}\left(p,q\right)$ of signature $\left(p,q\right)$ is the group of matrices that preserve the indefinite pseudo-Euclidean metric on the vector space $\mathbb{C}^{p,q}$. The goal of this paper is to describe the set ${\bf U}_{s}\left(p,p\right)$ of Hermitian, or, self-adjoint elements in ${\bf U}\left(p,p\right)$.