I need to generate random non-square matrices with $R$ rows and $C$ columns, elements randomly distributed with zero mean, and constrained such that the length ($L_2$ norm) of each row is $1$ and the length of each column is $\sqrt{\frac{R}{C}}$. Equivalently, the sum of square values is $1$ for each row and $\frac{R}{C}$ for each column.
So far I have found one way to achieve this: simply initialize the matrix elements randomly (e.g. from a uniform, normal, or laplace distribution with zero mean and arbitrary variance), then alternately normalize rows and columns to length $1$, ending with row normalization. This seems to converge to the desired result fairly quickly (e.g. for $R=40$ and $C=80$, variance of column length is typically ~ $~0.00001$ after $2$ iterations), but I'm not sure if I can depend on this fast convergence rate in general (for various matrix dimensions and initial element distributions).
My question is this: is there a way to achieve the desired result (row lengths $1$, column lengths $\sqrt{\frac{R}{C}}$) directly without iterating between row/column normalization? E.g. something like the algorithm for normalizing a random vector (initialize elements randomly, measure sum of square values, then scale each element by a common scalar). If not, is there a simple characterization for the convergence rate (e.g. num iterations until error $< \epsilon$) of the iterative method described above?
