The induced norm of the matrix $A$ as a map from $(\mathbb R^n , \| \cdot \|_p)$ to $(\mathbb R^n, \| \cdot \|_q)$ is given by $$ \| A \|_{p,q} = \sup_{x\in\mathbb{R}^n\setminus \{0\}} \frac{\|Ax\|_q}{\|x\|_p}.$$
I would like to compute this norm for $p,q\in\{1,2,\infty\}$. Thanks to Tom's answer to my previous question, the results for most of the cases can be found in the paper On the Calculation of the $l_2\to l_1$ Induced Matrix Norm.
The only case missing is $\| \cdot \|_{2,\infty}$. Is this still an open problem? 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.
You have, for $x$ with $\|x\|_2=1$ and using Cauchy-Schwarz, $$ \|Ax\|_\infty=\max_k\,\left|\sum_{j=1}^nA_{kj}x_j\right|\leq\max_k\left(\sum_{j=1}^n|A_{kj}|^2\right)^{1/2}=\max_k \|R_k(A)\|_2, $$ where $R_k(A)$ is the $k^{\rm th}$ row of $A$. Now suppose that we fix $k_0$ such that $\|R_{k_0}(A)\|_2$ is maximum among the rows. Let $x_0=R_{k_0}(A)/\|R_{k_0}(A)\|_2$. Then $$ \|Ax_0\|_\infty=\max_k\left|\sum_{j=1}^nA_{kj}\frac{A_{kj}}{\|R_k(A)\|_2}\right| =\max_k\frac{\|R_k(A)\|_2^2}{\|R_{k_0}(A)\|_2}=\|R_{k_0}(A)\|_2. $$ So $$ \|A\|_{2,\infty}=\max_k\|R_k(A)\|_2. $$