The geometry of the set of real square roots of $\pm I_2$

TL;DR

This paper explores the geometric structure of real square roots of ±I₂, revealing they form hyperboloids, and examines their algebraic forms using elementary methods, functions of matrices, and split-quaternions.

Contribution

It characterizes the geometry of square roots of ±I₂ as hyperboloids and introduces new algebraic descriptions using advanced mathematical tools.

Findings

Square roots of ±I₂ form hyperboloids of one or two sheets.

Involutory matrices of order 2 form a hyperboloid of one sheet.

Skew-involutory matrices of order 2 form a hyperboloid of two sheets.

Abstract

In this paper we study the geometry of the set of real square roots of $\pm I_2$. After some introductory remarks, we begin our study by deriving by quite elementary methods the forms of the real square roots of $\pm I_2$. We then discuss the interpretations of these square roots as transformations of the cartesian $(x,y)$-plane. To study the geometry of the set of square roots of $\pm I_2$ we consider a slightly more general set of square matrices of order $2$ and show that these sets are hyperboloids of one sheet or hyperboloids of two sheets. From these general results we conclude that the set of involutory matrices of order 2 is a hyperboloid of one sheet and the set of skew-involutory matrices of order 2 is a hyperboloid of two sheets. The relations between the geometrical properties of the hyperboloids and the set of square roots of $I_2$ are also investigated. We then proceed to obtain the forms of the involutory matrices of order 2 by more advanced methods. We have considered two approaches: in the first approach we use the concept of a function of a matrix and in the second approach we use concepts of split-quaternions.