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.

翻译：本文提出了一种生成随机正交矩阵 \(A\) 的算法方法，这些矩阵满足性质 \(A^t S A = S\)，其中 \(S\) 是一个固定的实可逆对称或斜对称矩阵。此方法具有重要意义，因为它推广了生成固定一般对称或斜对称双线性形式的正交矩阵的过程。这些矩阵包括属于诸如辛群、洛伦兹群、庞加莱群，以及更一般的不定正交群等群的正交矩阵。这些矩阵类别在多个领域发挥着关键作用，例如在理论物理中，它们被用于描述对称性和守恒定律；在计算几何、数值分析和数论中，它们则是研究二次型和模形式不可或缺的工具。我们的算法实现可以利用标准的线性代数库来完成。