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$A\in \mathbb{C}^{n\times n}$. Set: $$ r(A) = \max_{||u||_2=1}|u^* Au| $$

Prove the following statements:

(1) $||A||_2 \leq 2 r(A)$

(2)if $A^*A=AA^*$ then $r(A)=||A||_2$

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    The way this is phrased makes it sound like these are exercises from a book, or a course. If so, then ideally this should be acknowledged (at least in my personal opinion).2012-03-01

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