  We analyze the properties of entangled random pure states of a quantum system
partitioned into two smaller subsystems of dimensions $N$ and $M$. Framing the
problem in terms of random matrices with a fixed-trace constraint, we
establish, for arbitrary $N \leq M$, a general relation between the $n$-point
densities and the cross-moments of the eigenvalues of the reduced density
matrix, i.e. the so-called Schmidt eigenvalues, and the analogous functionals
of the eigenvalues of the Wishart-Laguerre ensemble of the random matrix
theory. This allows us to derive explicit expressions for two-level densities,
and also an exact expression for the variance of von Neumann entropy at finite
$N,M$. Then we focus on the moments $\mathbb{E}\{K^a\}$ of the Schmidt number
$K$, the reciprocal of the purity. This is a random variable supported on
$[1,N]$, which quantifies the number of degrees of freedom effectively
contributing to the entanglement. We derive a wealth of analytical results for
$\mathbb{E}\{K^a\}$ for $N = 2$ and $N=3$ and arbitrary $M$, and also for
square $N = M$ systems by spotting for the latter a connection with the
probability $P(x_{min}^{GUE} \geq \sqrt{2N}\xi)$ that the smallest eigenvalue
$x_{min}^{GUE}$ of a $N\times N$ matrix belonging to the Gaussian Unitary
Ensemble is larger than $\sqrt{2N}\xi$. As a byproduct, we present an exact
asymptotic expansion for $P(x_{min}^{GUE} \geq \sqrt{2N}\xi)$ for finite $N$ as
$\xi \to \infty$. Our results are corroborated by numerical simulations
whenever possible, with excellent agreement.
