I am curious about the current state of random matrix theory as pertains to the following question. I spent about an hour searching through literature and struggled to find material that directly answered this question, though I'm sure it's my own unfamiliarity with the field. I would be grateful for any reference recommendation.
Consider two random square matrices $A$ and $B$ which are iid.
For many continuous distributions, it is almost certain that they will be invertible.
Thus, consider the random matrix $B A^{-1}$.
I want to understand the reconstruction property of this random matrix $B A^{-1}$, namely
- how close is it to the identity matrix (e.g. are there bounds on $| A^{-1} B - I|$
- for some set $S$ of vectors, what is the average reconstruction error $|x - B A^{-1} x|$.
I'd be grateful for any and all results or references. I struggled to find pertinent results, even or perhaps especially under more specific versions of the above problem (e.g. each entry of $A$ and $B$ is an idd standard normal). Most of the results I saw focused more on bounding the operator norm or describing the eigenvalue distribution of a single random matrix or its inverse, but not the specific composition I describe, nor its reconstruction error.