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  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 $$n>1$$

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How are you defining your matrix norms? And, is this a homework problem? – Aaron May 11 '11 at 22:50
Isn't 1) the definition of the matrix p-norm? – Jose27 May 11 '11 at 22:52
@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. – Aaron May 11 '11 at 22:59

Instead of asking question in a hurry, you'd better write down what have you tried:-)

Before doing the proof, you may want to answer the following questions:

  • What's the definition of $\|A\|_p$? (and what is $p$ here?)

  • What do you know about $\|A\|_2$ and $\|A\|_F$?

After answering the questions above, you may be able to answer yours.

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Hint: $\|A\|_F^2=\sum_{j=1}^n \|a_j\|_2^2$ where $a_j$ is the vector given by the $j$th column of $A$. Can you show that $\|a_j\|_2\le \|A\|_2$?

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