We develop an efficient algorithm for sampling the eigenvalues of random matrices distributed according to the Haar measure over the orthogonal or unitary group. Our technique samples directly a factorization of the Hessenberg form of such matrices, and then computes their eigenvalues with a tailored core-chasing algorithm. This approach requires a number of floating-point operations that is quadratic in the order of the matrix being sampled, and can be adapted to other matrix groups. In particular, we explain how it can be used to sample the Haar measure over the special orthogonal and unitary groups and the conditional probability distribution obtained by requiring the determinant of the sampled matrix be a given complex number on the complex unit circle.

中文翻译：

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对随机正交矩阵和酉矩阵的特征值进行采样

我们开发了一种有效的算法，用于对根据正交或酉群上的 Haar 度量分布的随机矩阵的特征值进行采样。我们的技术直接对此类矩阵的 Hessenberg 形式的因式分解进行采样，然后使用定制的核心追踪算法计算它们的特征值。这种方法需要许多浮点运算，这些运算按照被采样的矩阵的顺序是二次的，并且可以适用于其他矩阵组。特别地，我们解释了如何使用它对特殊正交群和酉群上的 Haar 测度进行采样，以及通过要求采样矩阵的行列式是复单位圆上的给定复数而获得的条件概率分布。