Let $U$ be orthogonal. How can I prove that $||UA||_2=||A||_2$?

Question

I know that $||UA||_2\le||U||_2||A||_2$ and I also know that as $U$ is orthogonal, $U^{-1}=U^T$. But I don't know what else to do...

score 2 · Accepted Answer · answered Jun 14 '20 at 21:35

2

Hint: since $U^TU=I$, $(UA)^TUA=A^TU^TUA=\cdots$

answered Jun 14 '20 at 21:35

J.G.

Oh so, I know that $||UA||_2^2=(UA)^T UA=A^T A$ and $||A||_2^2=A^T A$. So can I say that if $||UA||_2^2 = ||A||_2^2$, is true that $||UA||_2 = ||A||_2$ ??? – User160 Jun 14 '20 at 21:45
@User160 You've glossed over one subtlety explained in the last comment here. To finish, take the square root. – J.G. Jun 14 '20 at 22:41

1 Answers1