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So, I know that $||AB|| \leq ||A||\cdot||B||$ (2-norm)

I'm doing a work on matrix algorithms and i seem to get as a result that $||A^TA|| = ||A||^2$

Does this always apply, or when and why does it happen?

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Which matrix norm are you using? –  Igor Rivin Dec 13 '11 at 14:05
    
2-norm, sorry for not mentioning that –  Koeneuze Dec 13 '11 at 14:15
    
I think this would be much better asked on math.stackexchange.com as it's not really research level. –  Matthew Daws Dec 13 '11 at 14:53
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...and, if you do ask at stackexchange, it would be better to give, explicitly, what your definition of the 2-norm is. –  Matthew Daws Dec 13 '11 at 14:59
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closed as too localized by Martin Brandenburg, Suvrit, Matthew Daws, Igor Rivin, Denis Serre Dec 13 '11 at 15:21

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1 Answer

If I'm not missing something, this is very obvious: $|A|^2_2$ is the maximal absolute value of the eigenvalues of $B = A^T A$. Thus $|A^TA|_2 =|B|_2$ is the square root of the maximal absolute value of the eigenvalues of $B^TB = B^2$. Since $B$ is diagonalizable, the eigenvalues of $B^2$ are just the squares of those of $B$.

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sorry, shouldn't this be a comment then? –  Suvrit Dec 13 '11 at 14:50
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looks like an answer to me... –  Igor Rivin Dec 13 '11 at 14:55
    
Yes, but I thought implicit MO etiquette suggested not answering blatantly non research questions, where comments would suffice. anyhow, not a big deal ;-) –  Suvrit Dec 13 '11 at 17:08
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