  Complex Hermitian random matrices with a unitary symmetry can be
distinguished by a weight function. When this is even, it is a known result
that the distribution of the singular values can be decomposed as the
superposition of two independent eigenvalue sequences distributed according to
particular matrix ensembles with chiral unitary symmetry. We give
decompositions of the distribution of singular values, and the decimation of
the singular values --- whereby only even, or odd, labels are observed --- for
real symmetric random matrices with an orthogonal symmetry, and even weight.
This requires further specifying the functional form of the weight to one of
three types --- Gauss, symmetric Jacobi or Cauchy. Inter-relations between gap
probabilities with orthogonal and unitary symmetry follow as a corollary. The
Gauss case has appeared in a recent work of Bornemann and La Croix. The Cauchy
case, when appropriately specialised and upon stereographic projection, gives
decompositions for the analogue of the singular values for the circular unitary
and circular orthogonal ensembles.
