TITLE: Is the max matrix norm induced?
QUESTION [5 upvotes]: Let $\|A \| = \max_{1 \le i,j \le n} |a_{ij}|$, where $A$ is a square matrix. I can prove that this is a matrix norm, but is it an induced norm?

REPLY [5 votes]: No. It's not an induced norm. Let $\|\cdot \|$ be your norm. If it were is an induced norm then you'd have $\|A ^2 \| \leq \|A\| ^ 2$ for every matrix $A$. But consider  $A = \left( \begin{array}{ccc}
1 & 1 \\
0 & 1 \end{array} \right)$ to get $2 = \|A ^2 \| \nleq \|A\| ^ 2 = 1^2 = 1$. Thus, your norm can't be an induced one.