Concentration of the smallest eigenvalue of a Sub-Gaussian matrix

Question

I would like to calculate the following probability:

$$\mathcal{P}(\lambda_{\min}(\mathbf{A})\geq t), \forall t\geq 0,$$

where $\lambda_{\min}$ is the smallest eigenvalue of $\mathbf{A}$ and each entry $\mathbf{A}_{r,s},\forall r,s \in \{1,\cdots,n\}$, is a subgassian random variable with parameter $v$.

Thank you for your help in advance.

Answer 1

I think the following Gordan's Theorem for Gaussian Matrices could be useful.

Let $\mathbf{A}$ be an $N\times n$ matrix whose entries are independent standard Gaussian random variables. Then $$\sqrt{N}-\sqrt{n} \leq \mathbb{E}\lambda_{\min}(\mathbf{A})\leq \mathbb{E}\lambda_{\max}(\mathbf{A}) \leq \sqrt{N}+\sqrt{n}$$

Well, when you have an estimate of expectation, you can adopt Markov inequality to estimate the concentration bound for $\mathcal{P} \left(\lambda_{\min}(\mathbf{A})\geq t\right), \forall t\geq 0$.

Your question is tightly connected with random matrices theory, and a very useful reference is following:

Vershynin R. Introduction to the non-asymptotic analysis of random matrices[J]. arXiv preprint arXiv:1011.3027, 2010.