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The Haar measure and the generation of random unitary matrices

This paper derives the Haar measure over the set of unitary matrices. The Haar measure is essential when studying the statistical behavior of complex sample covariance matrices in terms of their eigenvalues and eigenvectors. The characterization is based on Murnaghan's parameterization of unitary matrices which can be seen as a generalization of the representation of orthogonal matrices using Givens rotations. In addition to deriving the Haar measure, an efficient method to obtain samples from it is also presented.