The induced norm of the matrix $A$ as a map from $(\mathbb R^n , \| \cdot \|\_p)$ to $(\mathbb R^m, \| \cdot \|\_q)$ is given by
$$ \| A \|\_{p,q} = \sup\_{x\in\mathbb{R}^n\setminus \{0\}} \frac{\|Ax\|\_q}{\|x\|\_p}.$$


I would like to compute $\| \cdot \|\_{2,\infty}$. In this paper: [On the Calculation of the $l\_2\to l\_1$ Induced Matrix Norm](http://www.m-hikari.com/ija/ija-password-2009/ija-password5-8-2009/drakakisIJA5-8-2009.pdf), the authors presented the results for any $p,q\in\{1,2,\infty\}$, except for $p=2$ and $q=\infty$. 


So I would like to know if this is still an open problem. And if it is, then is there any result on finding a tight and easy-to-compute upper bound on the norm?


Thank you in advance.



On the induced matrix norm $\| \cdot \|_{2,\infty}$
Computing such induced norms is a hard problem. For the case of $p=2$ and $q \ge 2$, have a look at [this paper by Barak et al.](http://arxiv.org/abs/1205.4484) to see how tricky the problem is.


Typically, for other than the nice cases of $1,2, \infty$ style, these norms are NP-hard to compute, with well-known results for the case $p \ge q$ (see also e.g.: ["Matrix norms are NP-Hard to approximate"](http://dspace.mit.edu/openaccess-disseminate/1721.1/64956)). The paper of Barak et al focuses on the *hypercontractive* case of $p < q$.


For the specific case of $p=2$ and $q=\infty$ mentioned above, ~~I think~~ the paper of Barak et al cited above mentions that the norm is just the largest 2-norm of any row of the matrix (thanks to N. Johnston for pointing this out).


