  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 $\pm$1/2. Similarly, we write
the absolute value of the determinant of the $n\times{}n$ 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.
