Hi,

my book says that it is easy to show that for all orthogonal Q and Z of appropriate dimensions we have

$\displaystyle ||QAZ||_2 = ||A||_2$.

I'm thinking that $\displaystyle Q\in \mathbb{C}^{m\times m}$, $\displaystyle A\in \mathbb{C}^{m\times n}$ and $\displaystyle Z\in \mathbb{C}^{n\times n}$.

The 2-norm of a matrix is defined as,

$\displaystyle ||A||_2 = \sup_{||x||_2=1}||Ax||_2$

and so

$\displaystyle ||QAZ||_2 = \sup_{||x||_2=1}||(QAZ)x||_2$.

I do not know what to do here.. Any tips are greatly appreciated. Thanks.