  A feature of certain ensembles of random matrices is that the corresponding
measure is invariant under conjugation by unitary matrices. Study of such
ensembles realised by matrices with Gaussian entries leads to statistical
quantities related to the eigenspectrum, such as the distribution of the
largest eigenvalue, which can be expressed as multidimensional integrals or
equivalently as determinants. These distributions are well known to be
$\tau$-functions for Painlev\'e systems, allowing for the former to be
characterised as the solution of certain nonlinear equations. We consider the
random matrix ensembles for which the nonlinear equation is the $\sigma$ form
of \PVI. Known results are reviewed, as is their implication by way of series
expansions for the distributions. New results are given for the boundary
conditions in the neighbourhood of the fixed singularities at $t=0,1,\infty$ of
$\sigma$\PVI displayed by a generalisation of the generating function for the
distributions. The structure of these expansions is related to Jimbo's general
expansions for the $\tau$-function of $\sigma$\PVI in the neighbourhood of its
fixed singularities, and this theory is itself put in its context of the linear
isomonodromy problem relating to \PVI.
