Show that ‖AC‖2≤‖A‖2‖C‖2,
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‖≤sup‖x‖=1‖A‖‖Cx‖=‖A‖sup‖x‖=1‖Cx‖=‖A‖‖C‖.
Here we only used that
‖ACx‖≤‖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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