What is the explicit expression of the operator norm of $A$ : $(\mathbb{R}^n, |\cdot |_1) \rightarrow (\mathbb{R}^m, |\cdot |_{\infty}) $

Question

What is the explicit expression of the operator norm of $A$ : $(\mathbb{R}^n, |\cdot |_1) \rightarrow (\mathbb{R}^m, |\cdot |_{\infty}) $ ?

I have no idea that what is "explicit expression" means, and what I should work on it? All I know is just the definition of matrix norm...

$$ \max_{|x|=1} |Ax|_{\infty} = \max_i \sum^n_{j=1} |a_{ij}x_{ij}| = \max_i \sum^n_{j=1} |a_{ij}| |x_{ij}| $$

Answer 1

For $y = Ax$ you have $$ \| y \|_\infty = \max_j \left| \sum_{k=1}^n a_{jk} x_k \right| \le \max_j \sum_{k=1}^n |a_{jk} | \, | x_k | \\ \le \left(\max_{j, k} |a_{jk} | \right) \sum_{k=1}^n | x_k | = \left(\max_{j, k} |a_{jk} | \right) \| x \|_1 \tag 1 $$ It follows that for the norm $\|A\|$ of the operator $A : (\mathbb{R}^n, \|\cdot \|_1) \rightarrow (\mathbb{R}^m, \|\cdot |\|_{\infty})$ you have $$ \|A\| = \max_{ \| x \|_1 = 1} \| Ax \|_\infty \le \max_{j, k} |a_{jk} | \, . \tag 2 $$ Now $ \max_{j, k} |a_{jk} | = |a_{J, K}|$ for some index pair $(J,K)$. For the vector $$ x = (0, \dots, 0, 1, 0, \dots, 0)^T $$ which has a one at the $K^\text{th}$ component, equality holds in $(1)$ and therefore also in $(2)$. So $\|A\| = \max_{j, k} |a_{jk} |$, which is an "explicit expression" of the operator norm in terms of the elements of the matrix $A$.