Suppose I sample $m\times n$ matrix $X$ with IID entries sampled from standard Gaussian and $m,n$ are both large. What is the distribution of $k$th squared singular value?
I simulated singular values from some 100x100 random matrices below. Means of these seem to be well predicted by Marchenko-Pastur distribution with shape parameter 1. What about the variances?
Looks like for the top eigenvalue, distribution approaches Tracy-Widom (notebook)