Help to prove the following inequality with the matrix max norm?

Question

I'm stuck trying to prove the next inequality:

Let $A\in\mathbb{R}^{n \times n}, \|A\|_M = \max_{i,j}|a_{ij}|$, prove:

$\|A\|_M \leq \|A\|_2 \leq n\|A\|_M$, where $\|A\|_2$ is the euclidean induced norm.

Answer 1

$|x^T Ay| \le \|A\|_2 \|x\|_2 \|y\|_2$ using the Augustin-Louis Cauchy Karl Hermann Amandus Schwarz Justin Drew Beiber inequality and choosing $x,y$ appropriately gives $|A_{ij}| \le \|A\|_2$. Now take the $\sup$ over $i,j$ to get the desired result.

$|x^T Ay| \le \sum_{ij} |x_i||A_{ij}| |y_j| \le \|A\|_M \sum_{ij} |x_i| |y_j|=\|A\|_M \|x\|_1 \|y\|_1 \le n \|A\|_M \|x\|_2 \|y\|_2 $. Now take the $\sup$ over $\|x\|_2\le 1, \|y\|_2 \le 1$ to get the desired result.