## The singular values of the GUE (less is more)

##### Author(s)

Edelman, Alan; La Croix, Michael Andrew
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Some properties that nominally involve the eigenvalues of Gaussian Unitary
Ensemble (GUE) can instead be phrased in terms of singular values. By
discarding the signs of the eigenvalues, we gain access to a surprising
decomposition: the singular values of the GUE are distributed as the union of the singular values of two independent ensembles of Laguerre type. This
independence is remarkable given the well known phenomenon of eigenvalue repulsion. The structure of this decomposition reveals that several existing observations about large n limits of the GUE are in fact manifestations of phenomena that are already present for finite random matrices. We relate the semicircle law to the quarter-circle law by connecting Hermite polynomials to generalized Laguerre polynomials with parameter ± 1/2. Similarly, we write the absolute value of the determinant of the n x n GUE as a product n independent random variables to gain new insight into its asymptotic log-normality. The decomposition also provides a description of the distribution of the smallest singular value of the GUE, which in turn permits the study of the leading order behavior of the condition number of GUE matrices. The study is motivated by questions involving the enumeration of orientable maps, and is related to questions involving powers of complex Ginibre matrices. The inescapable conclusion of this work is that the singular values of the GUE play an unpredictably important role that had gone unnoticed for decades even though, in hindsight, so many clues had been around.

##### Date issued

2015-10##### Department

Massachusetts Institute of Technology. Department of Mathematics##### Journal

Random Matrices: Theory and Applications

##### Publisher

World Scientific

##### Citation

Edelman, Alan, and Michael La Croix. “The Singular Values of the GUE (less Is More).” Random Matrices: Theory and Applications 4, 4 (October 2015): 1550021

Version: Original manuscript

##### ISSN

2010-3263

2010-3271