On the singular values of Gaussian random matrices

Abstract

This short note is about the singular value distribution of Gaussian random matrices (i.e. Gaussian Ensemble or GE) of size N. We present a new approach for deriving the p.d.f. of the singular values directly from the singular value decomposition (SVD) form, which also takes advantage of the rotational invariance of GE and the Lie algebra of the orthogonal group. Our method is direct and more general than the conventional approach that relies on the Wishart Ensemble and the combination of QR and Cholesky decomposition. Directly based on this p.d.f., and its interpretation by statistical mechanics, we give the physics proof that in the thermodynamic limit ( N→∞), the singular value distribution satisfies the quadrant law, similar to the celebrated semi-circle law established by Wigner more than 40 years ago for the spectral distribution of Gaussian Orthogonal (or Unitary) Ensembles. This quadrant law was also proved earlier and mathematically more rigorously by some authors based on probabilistic estimations and the moment method, but not directly from the p.d.f. formula.