Jensen Inequality Euclidean Norm

Question

I'm trying to prove $$\mathbb{E}\|X\|\geq \|\mathbb EX\| $$ for any random matrix $X$ and the spectral norm $\|\cdot \|$. To finish my proof all I need is the same statement for the euclidean norm $|\cdot|$ of a random vector $v$: $$\mathbb{E}|v|\geq |\mathbb Ev|$$ But I am stuck trying to prove this. I am familiar with the one-dimensional Jensen inequality. Am I simply missing the right application of it here?

Answer 1

For any unit vector $u$ we have $\mathbb E |v| \geq \mathbb E \langle u,v \rangle = \langle u,\mathbb Ev \rangle $. The supremum of the right side over all unit vectors $u$ is equal to $|\mathbb E v|$.