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.