  As a unifying framework for examining several properties that nominally
involve eigenvalues, we present a particular structure of the singular values
of the Gaussian orthogonal ensemble (GOE): the even-location singular values
are distributed as the positive eigenvalues of a Gaussian ensemble with chiral
unitary symmetry (anti-GUE), while the odd-location singular values,
conditioned on the even-location ones, can be algebraically transformed into a
set of independent $\chi$-distributed random variables. We discuss three
applications of this structure: first, there is a pair of bidiagonal square
matrices, whose singular values are jointly distributed as the even- and
odd-location ones of the GOE; second, the magnitude of the determinant of the
GOE is distributed as a product of simple independent random variables; third,
on symmetric intervals, the gap probabilities of the GOE can be expressed in
terms of the Laguerre unitary ensemble (LUE). We work specifically with
matrices of finite order, but by passing to a large matrix limit, we also
obtain new insight into asymptotic properties such as the central limit theorem
of the determinant or the gap probabilities in the bulk-scaling limit. The
analysis in this paper avoids much of the technical machinery (e.g. Pfaffians,
skew-orthogonal polynomials, martingales, Meijer $G$-function, etc.) that was
previously used to analyze some of the applications.
