Saturday, 5 November 2016

linear algebra - Different version of Cauchy-Schwarz Inequality?



Show that



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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