  Rectangular real $N \times (N + \nu)$ matrices $W$ with a Gaussian
distribution appear very frequently in data analysis, condensed matter physics
and quantum field theory. A central question concerns the correlations encoded
in the spectral statistics of $WW^T$. The extreme eigenvalues of $W W^T$ are of
particular interest. We explicitly compute the distribution and the gap
probability of the smallest non-zero eigenvalue in this ensemble, both for
arbitrary fixed $N$ and $\nu$, and in the universal large $N$ limit with $\nu$
fixed. We uncover an integrable Pfaffian structure valid for all even values of
$\nu\geq 0$. This extends previous results for odd $\nu$ at infinite $N$ and
recursive results for finite $N$ and for all $\nu$. Our mathematical results
include the computation of expectation values of half integer powers of
characteristic polynomials.
