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.
Best Answer
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 $\|Mx\|_\infty = C$.