Characterization of spectral norm: $\left\Vert A\right\Vert _{2}=\sup\left\{ \text{trace}\left(ZA\right)\ :\ Z\succeq0\right\}$

Question

I came across a claim that given $A\in\mathbb{R}^{n\times n}$ the following equality holds $$\left\Vert A\right\Vert _{2}=\sup\left\{ \text{trace}\left(ZA\right)\ :\ Z\succeq0\right\}$$ Where $\left\Vert A\right\Vert _{2}=\sup_{\left\Vert x\right\Vert _{2}=1}\left\Vert Ax\right\Vert _{2}$ w.r.t the Euclidian norm on $\mathbb{R}^{n}$. I can't figure out whether this is actually correct, if it is I would be happy to see a proof.

Answer 1

Do you have some restrictions on $Z$? otherwise if we take $Z$ to be the identity matrix, then your proposition would say $||A||_2 \geq Trace(A)$ which is untrue.