The eigenvalue density

Write $\alpha =n-m$. The squared singular values $x_1,\dots ,x_m$ of $G$ have joint density proportional to

For completeness, the singular-value change of variables gives one factor ${x_i}^{n-m}$ from the $n-m$ unused column directions for each singular value. The two complex angular directions associated with a pair of distinct singular values give $(x_i-x_j{)}^2$. The radial coordinates contribute $d{x}_i$. Integrating the angular variables leaves a constant independent of the $x_i$. Repeated singular values form a set of Gaussian measure zero, so this change of variables determines the density.

Define generalized Laguerre polynomials by Rodrigues's formula

Integration by parts gives orthogonality with weight $w_{\alpha }(x)=e^{-x}{x}^{\alpha }$, and norm

Indeed, ${L_j}^{\alpha }$ has leading coefficient $(-1{)}^j/j!$; it is orthogonal to every lower-degree polynomial, and integrating it against $x^j$ gives $(-1{)}^j\Gamma (j+\alpha +1)$.

Replacing the monomials in the Vandermonde determinant by these polynomials changes only a nonzero constant. Expanding the squared determinant and integrating with orthogonality shows that the one-point eigenvalue density is

Its integral is $m$. Therefore