Saturday, 5 November 2016

linear algebra - Different version of Cauchy-Schwarz Inequality?



Show that AC2A2C2,



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=supx=1ACxsupx=1ACx=Asupx=1Cx=AC.


Here we only used that
ACxACx


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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