0
$\begingroup$
  1. $$\|A\|_p = \displaystyle \max_{\|x\|_p = 1} \|Ax\|_p $$
  2. $$\|A\|_2 \leq \|A\|_F \leq \sqrt{n}\|A\|_2$$

How I can show that $1$ and $2$ are correct?

$2)$ $||Ax||_{2}=\sqrt{\sum_{i=1}^{n} | \sum_{j=1}^{n} a_{kj},x_{j}|^{2}}\leq \sqrt{[\sum_{k=1}^{n}(\sum_{j=1}^{n} |a_{kj}|^2)(\sum_{j=1}|x_{j}|^{2})]} = ||A||_{F} ||x||_{2} $

but how i show $$||A||_{F}\leq \sqrt{n}||A||_{2}$$

$||A||_{F} \leq (\sqrt{n*\sum_{i=1}^{n} | \sum_{j=1}^{n} a_{kj},x_{j}|^{2}})$

with $$n>1$$

  • 2
    How are you defining your matrix norms? And, is this a homework problem?2011-05-11
  • 1
    Isn't 1) the definition of the matrix p-norm?2011-05-11
  • 1
    @Jose27: If he's asking to show that it is correct, it couldn't be the definition. It is definitely a possible definition, but he has to be starting from some other point.2011-05-11

2 Answers 2