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where A, C are square matrices of the same size.
I understand this probably needs to use Cauchy-Schwarz Inequality, but I don't know how should I start the problem, any help?
Answer
Observe that
\|AC\|=\sup_{\|x\|=1}\|ACx\|\le \sup_{\|x\|=1}\|A\|\|Cx\|=\|A\|\sup_{\|x\|=1}\|Cx\|=\|A\|\|C\|.
Here we only used that
\|ACx\|\le \|A\|\|Cx\|
which is an implication of the definition of \|A\|.
Note. This inequality HAS NOTHING TO DO with Cauchy-Schwarz, although it looks the same!
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