I would be really thankful if you help me solve this problem. I need to prove that if $P$ and $Q$ are orthogonal matrices, this is true: $$\mathbf{\|PAQ\|_2=\|A\|_2}$$ I've only got to this : $$\|\mathbf{PAQ}\|_2=\max\limits_{||x||_2=1}\|\mathbf{PAQ}x\|_2 $$ As $\mathbf{P}$ is an orthogonal matrix, $$\max\limits_{ \|x||_2=1}\|\mathbf{PAQ}x\|_2=\max\limits_{||x||_2=1}\|\mathbf{AQ}x\|_2$$. I'm not sure about how to continue. Thanks!!!!
2026-04-08 06:34:42.1775630082
Orthogonal matrix: 2 norm
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Note, $x$ ranges over the unit ball and not the whole space.
Observe that $\{Qx:\|x\|\le1\} =\{y:\|y\|\le 1\}$, that is, $Q$ maps the unit ball into itself, bijectively because we're in a finite dimensional space. So, their supremum is the same.