What is the closed form for this norm $\|Z\|=\sup_{\|u\|=1,\|v\|=1} u^TZv$?

Question

I read a paper, which has the equation above $\|Z\|=\sup_{\|u\|=1,\|v\|=1} u^TZv$. Is this a well known norm and what is the closed form for this norm. Also what does this norm describe?

Thanks in advance.

Answer 1

If $\|\cdot\|$ denotes the euclidian norm, for $Z\in\mathbb{R}^{n\times n}\setminus \{0\}$ we have \begin{gather} \sup_{\|v\|=1} \|Zv\|=\sup_{\|v\|=1,Zv\neq0} \frac{\|Zv\|^2}{\|Zv\|}= \sup_{\|v\|=1,Zv\neq0} (\frac{Zv}{\|Zv\|})^{\top} (Zv)\\ \leq \sup_{\|u|=1,\|v\|=1}u^\top (Zv)\stackrel{\mathrm{csi}}{\leq}\sup_{\|u|=1,\|v\|=1}\|u\|*\|Zv\| =\sup_{\|v\|=1}\|Zv\|. \end{gather}

This implies \begin{gather}\sup_{\|u|=1,\|v\|=1}u^\top Zv=\sup_{\|v\|=1} \|Zv\|,\end{gather} so your given norm is the spectral norm.