Let $M:\mathbb{C}\to \mathbb{C}$ be a matrix and equip $\mathbb{C}$ with the norm $$\|x\|_\infty=\max_{1\le j\le n}|x_j|.$$ If the operator norm is given by $$\|M\|=\sup_{\|x\|=1}|Mx|,$$ is it possible to compute the operator norm exactly in terms of the matrix entries?
Since $(Mx)_i=\sum_{j=1}^nm_{ij}x_j$, we have $$\|M\|=\sup_{\|x\|_\infty=1}\max_{1\le i\le n}\bigg|\sum_{j=1}^nm_{ij}x_j\bigg|.$$ From here, it is clear how one might bound this norm, but it is not clear to me how to compute it exactly without knowledge of the matrix.
Hint: Using the triangle inequality, show that if $\|x\|_\infty = 1$, then $$ |(Mx)_i| \leq \sum_{j=1}^n |m_{ij}|. $$ This gives you an upper bound for $\|M\|$, i.e. a value $C$ that depends on the entries of $M$ for which $\|M\| \leq C$. Using the entries of $M$, find a vector $x$ for which $\|x\|_\infty = 1$ and $\|Mx\|_\infty = C$.