0
$\begingroup$

Let $A$ be a random $m$-by-$n$ matrix with iid $N(0,1)$ entries, $m$ and $n$ large with $n/m \longrightarrow \alpha \in (0, 1)$ . Let $\|A\|_2$ be the largest singular value of $A$ (i.e the spectral norm of $A$) and let $t \ge 0$.

Question. What is a good upper bound for $\mathbb E_A[e^{-t\|A\|_2}]$ ?

Improve this question
$\endgroup$
1
  • $\begingroup$ I've attempted to make the title identical to the body. I've also added details about the size of shape of the matrix. Hope you're fine with the current version. Thanks. $\endgroup$
    – dohmatob
    Commented Apr 25, 2020 at 14:56

1 Answer 1

Reset to default
1
$\begingroup$

The probability distribution of the largest singular value of $A$ (or the largest eigenvalue $\lambda_1$ of $AA^T$) is derived in Distribution of the largest eigenvalue for real Wishart and Gaussian random matrices and a simple approximation for the Tracy–Widom distribution. There are exact answers for finite $n,m$ and asymptotic forms for large $n,m$. From here finding the desired average of $e^{-t\lambda_1}$ is a matter of quadrature.

Improve this answer
$\endgroup$
1
  • $\begingroup$ Great. Thanks for the reference. At the moment, I also came across this reference arxiv.org/pdf/1003.2990.pdf. A little twist: largest singular value of A is the square root of the largest eigenvalue of $AA^T$. $\endgroup$
    – dohmatob
    Commented Apr 25, 2020 at 15:02

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .