Generation of Random (Generalized) Orthogonal Matrices

TL;DR

This paper introduces a versatile algorithm for generating random orthogonal matrices that preserve a given bilinear form, encompassing various important groups like symplectic and Lorentz groups, with applications across physics and mathematics.

Contribution

It provides a generalized method for generating orthogonal matrices that fix arbitrary symmetric or skew-symmetric forms, extending existing procedures to broader matrix groups.

Findings

Algorithm can generate matrices for symplectic, Lorentz, and Poincaré groups.

Method is implementable with standard linear algebra libraries.

Applicable in physics, geometry, and number theory contexts.

Abstract

This paper presents an algorithmic method for generating random orthogonal matrices \(A\) that satisfy the property \(A^t S A = S\), where \(S\) is a fixed real invertible symmetric or skew-symmetric matrix. This method is significant as it generalizes the procedures for generating orthogonal matrices that fix a general fixed symmetric or skew-symmetric bilinear form. These include orthogonal matrices that fall to groups such as the symplectic group, Lorentz group, Poincar\'e group, and more generally the indefinite orthogonal group, to name a few. These classes of matrices play crucial roles in diverse fields such as theoretical physics, where they are used to describe symmetries and conservation laws, as well as in computational geometry, numerical analysis, and number theory, where they are integral to the study of quadratic forms and modular forms. The implementation of our algorithms can be accomplished using standard linear algebra libraries.