$l_2 \rightarrow l_\infty$ induced matrix norm

Question

I would like to know what is the induced matrix norm of a matrix $A$, when the domain space is equipped with $l_2$ norm and the range space is equipped with $l_\infty$ norm, i.e., what is $\lvert| A \rvert|_{2 \infty}$? The expression for $\lvert| A \rvert|_{\infty 2}$ is given in Konstantinos Drakakis - On the Calculation of the L2 to L1 Induced Matrix Norm.

Answer 1

In general, when $X$ is a normed space, the $X\to \ell_\infty$ norm of a matrix $A$ is computed as
$$ \sup_{\|x\|_X\le 1}\|Ax\|_\infty = \sup_{\|x\|_X\le 1} \max_i \left|\sum_{j} a_{ij}x_j\right| = \max_i \sup_{\|x\|_X\le 1} \left|\sum_{j} a_{ij}x_j\right| = \max_i \|A_{i*}\|_{X^*} $$ where $A_{i*}$ is the $i$th row of $A$, and $X^*$ is the dual of $X$. (Since $A_{i*}$ is a map from $X$ to scalars, its $X^*$ norm makes sense.) Simply put, when the target is $\ell_\infty$, the rows (components of the map) can be considered independently of one another.

In the special case $\ell_2\to\ell_\infty$ we end up with $$ \|A\|_{2\infty} = \max_i \sqrt{\sum_j |a_{ij}|^2} $$ which is the maximal Euclidean length of the rows of $A$.