1
$\begingroup$

Let $X$ be a random (symmetric) matrix drawn from an unknown distribution. I have an estimate of $\lambda_{\min}(\mathbf{E}[X])$. Specifically, I have $$\lambda_{\min}(\mathbf{E}[X]) \geq c$$ a constant $c$ which is very large.

Can we say anything about $$\mathbf{E}[\lambda_{\min}(X)]?$$ I know from Jensen's Inequality about concavity, that $$\mathbf{E}[\lambda_{\min}(X)] \leq \lambda_{\min}(\mathbf{E}[X]).$$

But that doesn't help me in saying $$\mathbf{E}[\lambda_{\min}(X)] \geq c.$$ My question is can the Jensen gap be so bad in such a situation that $\mathbf{E}[\lambda_{\min}(X)]$ is close to $0$ whereas $\lambda_{\min}(\mathbf{E}[X])$ is arbitrarily large?

Improve this question
$\endgroup$
2
  • $\begingroup$ simultaneously posted at math.stackexchange.com/questions/4249940/… $\endgroup$
    – Sean Eberhard
    Commented Sep 14, 2021 at 8:57
  • 1
    $\begingroup$ Let $X=\frac{n}{n-1} c(I - e_{k,k})$ where $e_{k,k}=1$ at the $k$th row and column and $0$ otherwise, and $k$ is a uniform random variable from $\{1,...,n\}$. Thus $X$ always have $0$ as an eigenvalue but $E[X]=cI$. $\endgroup$
    – LeechLattice
    Commented Sep 14, 2021 at 9:41

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .