Given $A \in R^{m\times n}$, I need to prove:
$||A||_2 \le \sqrt {m}||A||_\infty$
I have tried a number of things and I just cant seem to get it to work.
Also, I need to prove:
$||A||_2 \le \sqrt {n} ||A||_1$
For this one, I have done: $e_i = [...,0,1,0,...] \in R^n$ where i is the position of the 1 in e.
$||A||_1 = \max {\sum {|a_{ij}|}}=\max {||Ae_i||_1}$ $B => b_i = ||Ae_i||_1 \in R^n$ $||A||_1 = ||B||_\infty \le ||B||_2$ $||B||_2 = \sqrt {\sum {||Ae_i||_1^2}} \le \sqrt {\sum {||A||_1^2||e_i||_1^2}} = \sqrt {n}||A||_1$
I dont know if this is correct or not because I have no idea how to get this to be greater than or equal to $||A||_2$. I feel like this is very close to what I need, just not quite there. Any help would be great.