Generation of Random (Generalized) Orthogonal Matrices

TL;DR

The paper addresses sampling random matrices A that preserve a fixed nondegenerate bilinear form S by generating elements of the generalized orthogonal group O_S(N) ∩ O(N), where A^T S A = S and S is either symmetric or skew-symmetric. It leverages standard numerical linear algebra tools—QR factorization, real/complex Schur decompositions, and polar decomposition—to transform the problem into block-diagonal commuting constructions, yielding explicit, implementable algorithms. For symmetric S, it reduces to constructing a block-diagonal orthogonal matrix B that commutes with a diagonalized form T, then recovering A via A=U B U^T. For skew-symmetric S, it uses the real Schur form to reduce to a problem on a canonical J form and employs a unitary-to-real mapping to generate A from commuting unitary blocks; the method covers the Lorentz, symplectic, and more general indefinite orthogonal cases. The results provide practical procedures, compatible with standard linear algebra libraries, to sample generalized isometries with broad applications in physics, geometry, and number theory.

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, Poincaré 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.