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Thanks to the rotation-invariance of the Gaussian distribution, both $U_m$ and $V_m$ can be taken to be Haar-distributed orthogonal matrices in $\mathbb{R}^{m\times m}$ and $\mathbb{R}^{n\times n}$ re …

Carlo's answer addresses the first question. To address the second one: the "disk law" (better known as the "circular law") does not tell you the distribution of singular values. However, the Marchen …

I don't think there's an exact expression, but the Bai–Yin result does give the right prediction. It's a little easier to state nice-looking results for a $p \times n$ matrix $X$ with independent sta …

One way of looking at this is that, up to normalization, the GUE is standard Gaussian measure on the space of Hermitian matrices, equipped with the Hilbert-Schmidt or Frobenius inner product. So what …

What counts as "best"? The smallest tail bound is of course $$ (2\pi)^{-n/2} \int_{\{(x_1,\dotsc,x_n) : \sum x_i^4 > (1+t)3n\} } e^{-\sum x_i^2 / 2} dx_1 \dotsb dx_n. $$ Presumably you want somethin …

I don't have time for details right now, but you want to look at circa 1990 papers by Alan Edelman and Stanislaw Szarek (independently) on condition numbers of random matrices. For the state of the a …

I'm not sure offhand (and don't have time to check at the moment) if the GOE version of this is known, but the distribution least singular value of a nonsymmetric $n \times n$ matrix with i.i.d. norma …

There is active interest in such results in high-dimensional geometry, and expander graphs have even been used explicitly as a tool. Take a look for example at this paper and the references on the se …

Here's just one quick remark (a little involved for a comment) about how you can start making greater use of your suggested reduction to $A$. Since $M-A$ has rank 1, $$ \sigma_3(A) \le \sigma_2(M) \le …

As Terry pointed out in a comment, the TW law is only valid asymptotically. However, there are various results which show that the asymptotics for finite dimension, which TW heuristically suggest, ar …

I'm guessing that you meant to write "real orthogonal matrices" without "symmetric", since the set of symmetric orthogonal matrices has Haar measure 0. Otherwise, please clarify what you mean by "a d …

Here are some papers you might want to look at if you're interested in this issue (probably better than the paper of mine that Suvrit mentioned): G. Bennett, V. Goodman, and C.M. Newman. Norms of ra …

Assuming I'm interpreting your notation correctly, with high probability it is known that $\lambda_{\min}(W_n) \ge c/n$ for some absolute constant $c$ (see Edelman, "Eigenvalues and condition numbers …