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We develop an efficient algorithm for sampling the eigenvalues of randommatrices distributed according to the Haar measure over the orthogonal orunitary group. Our technique samples directly a factorization of the Hessenbergform of such matrices, and then computes their eigenvalues with a tailoredcore-chasing algorithm. This approach requires a number of floating-pointoperations that is quadratic in the order of the matrix being sampled, and canbe adapted to other matrix groups. In particular, we explain how it can be usedto sample the Haar measure over the special orthogonal and unitary groups andthe conditional probability distribution obtained by requiring the determinantof the sampled matrix be a given complex number on the complex unit circle.

Sampling the eigenvalues of random orthogonal and unitary matrices