Prove that orthogonal matrix $Q$ preserves operator norm i.e. $||AQ||_2 = ||A||_2$

Question

This question proves that $||QA||_2 = ||A||_2$:

$||QA||_2 = \sup_{||x||=1}{||QAx||}=\sup_{||x||=1}{\sqrt{(QAx)^T(QAx)}} = \sup_{||x||=1}{\sqrt{x^TA^TQ^TQAx|}} = sup_{||x||=1}{\sqrt{x^TA^TAx}} = ||A||_2$

How to prove it for $||AQ||_2$ too?

Answer 1

We denote $\langle x, y\rangle_2 = x^Ty$. Then you have $$ \Vert Qx \Vert_2 = \langle Qx, Qx\rangle_2 = \langle x, Q^TQx\rangle_2 = \langle x, x\rangle_2 = \Vert x \Vert_2.$$ Therefore we have $\{x \in \mathbb R^n : \Vert x \Vert_2 = 1\} = \{x \in \mathbb R^n : \Vert Qx \Vert_2 = 1\}$. Thus we get $\sup_{\Vert x \Vert_2 = 1}{\Vert AQx \Vert_2}= \sup_{\Vert Qx \Vert_2 = 1}{\Vert A Qx \Vert_2}$ for the suprema.

Hence you get $$\Vert AQ \Vert_2 = \sup_{\Vert x \Vert_2 = 1}{\Vert AQx \Vert_2}= \sup_{\Vert Qx \Vert_2 = 1}{\Vert A Qx \Vert_2} = \sup_{\Vert y \Vert_2 = 1}{\Vert Ay \Vert_2} = \Vert A \Vert_2.$$

Answer 2

The map $Q$ preserves the unit sphere, so the sup of $\|AQx\|$ over the unit sphere will be equal to the sup of $\|A\|$ over the unit sphere.