Lets say we are given random (row) vector $x \in \mathbb{R}^n$, and a non-random matrix $M \in \mathbb{R}_{nxn}$
I came across a claim (which does not impose any assumptions on the distribution of $x$ a priori), that the following expectation equality holds:
$$ \mathop{\mathbb{E}}[||x^tM||] = \sigma_x||M||_F^2$$
Where $||M||_F$ is the Frobenius matrix norm and $\sigma_x$ is the standard deviation of $x$ (identically component-wise I assume although again this was not qualified in the original statement)
I'm trying to understand this statement and prove it for myself. To begin with, trying it out on a 1x1 case I think there's an error in the claim in that it should use $||M||_F$ rather than $||M||_F^2$, is that right?
After that, I am still unclear how to derive the full equality. I was able to derive it as an inequality using Jensen's inequality and assuming the mean of $x$ is zero, but thats as far as I got.
Any insight appreciated on whether this statement is generally true. Alternatively, does it hold if we impose additional requirements on the vector $v$? e.g. zero-mean or stronger claims (such as component-wise i.i.d $\mathcal{N}(0,\,\sigma^{2})$ etc?)
