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

Abstract

The pseudo-unitary group $\mathbf{U}\left(p,q\right)$ 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 ${\mathbf{U}}_s\left(p,p\right)$ of Hermitian, or, self-adjoint elements in $\mathbf{U}\left(p,p\right)$.