  The distribution function for the first eigenvalue spacing in the Laguerre
unitary ensemble of finite rank random matrices is found in terms of a
Painlev\'e V system, and the solution of its associated linear isomonodromic
system. In particular it is characterised by the polynomial solutions to the
isomonodromic equations which are also orthogonal with respect to a deformation
of the Laguerre weight. In the scaling to the hard edge regime we find an
analogous situation where a certain Painlev\'e \IIId system and its associated
linear isomonodromic system characterise the scaled distribution. We undertake
extensive analytical studies of this system and use this knowledge to
accurately compute the distribution and its moments for various values of the
parameter $ a $. In particular choosing $ a=\pm 1/2 $ allows the first
eigenvalue spacing distribution for random real orthogonal matrices to be
computed.
