Distribution of singular values of a random matrix with columns constrained to be restrictly positive or negative.

Question

Let $X_n$ be a $n \times n$ real matrix, and all extries expect its last column are independently sampled from an exponential distribution $exp(1)$; the entries of the last column are independently sampled from another exponential distribution $exp(\frac{1}{n-1})$ and then multiply by -1. Therefore, now we have a random matrix with column sign constraint, and the expectation of the sum of entries $\mathbb{E}[x]=0$. How will the distribution of its singular values behave? I tried to do some simulation ($n=100$) and it seems there will be two parts: one following marchenko-pastur law, and the other part is just a single spike. Is there any proof of this? Thanks in advance!

Answer 1

Using $n=100$ and 1,000 simulations with Mathematica, I get the following for the distribution of the log of the $1000n$ singular values:

SeedRandom[12345];
nsim = 1000;
data = {};
n = 100;
Do[
  (* Generate matrix *)
  m = Table[If[j == n, -RandomVariate[ExponentialDistribution[1/(n - 1)], 1][[1]],
    RandomVariate[ExponentialDistribution[1], 1][[1]]], {i, n}, {j, n}];

  (* Get singular values *)
  sv = SingularValueList[m];

  (* Add to data *)
  data = Join[data, sv],
{k, nsim}]

MinMax[data]
(* {0.0000822545, 1942.99} *)

Histogram[Log[data], "FreedmanDiaconis", "PDF", PlotRange -> {{-8, 8}, {0, 0.62}},
  Frame -> True, PlotLabel -> Style["n = " <> ToString[n], Bold, 24],
  FrameLabel -> (Style[#, Bold, 18] &) /@ {"log(singular value)", "Probability density"}]

For $n=100$, there appears to be two small "spikes" to the right of most of the distribution of the log of singular values.

One can use the above code to generate an animation of the changes in distribution as $n$ goes from 2 to 50 (or more):