Recently i got stuck into a problem and couldn't find its satisfactory answer anywhere.

My question is simple. Suppose i have a fat random matrix (i,e $R$ has dimensions $k\times d$ where $k<d$) whose
elements are chosen from a i.i.d standard normal distribution N(0,1).

If i find its pseudo inverse, given by: $R^+ = (R' R)^{-1} R'$.

- Will the this pseudo inverse matrix will still remain random ?
- If yes, will it contain elements distributed with normal distribution?
- If yes, what would be the mean and variance of this this normal distribution?

I am asking these questions because i have experimented with a lot of random matrices (with elements distributed with N(0,1). When in plot a histogram of pseudo inverse elements, it comes a normal distribution with mean = 0 and variance = 1/(variance of $R$ $\times d^2$) ; where d are the columns in R.)

I have tried to find PDF using Jacobian transform but i could not figure out how will it shape up the variance.

I would be thankful if you could guide me or clarify my problem. Thanks,