Proof that the 2-norm of orthogonal transformation of a matrix is invariant

Question

For any matrix A and an orthogonal matrix Q, I can prove in the standard way that $$||QA||_2 = ||A||_2 $$ using $$||QA||_2^2 = (QAx)^T(QAx) = (Ax)^T(Ax) = ||A||_2^2 $$

However, I am unable to cancel Q, when the transformation is AQ, i.e. $$||AQ||_2^2 = (AQx)^T(AQx) = x^TQ^TA^TAQx$$ After this point I am unable to prove the same that $$||AQ||_2 = ||A||_2 $$

Answer 1

Recall that the 2-norm for matrices is defined as

$$ ||B||_2 \;\; =\;\; \sup_{||x||=1} ||Bx||. $$

But for any orthogonal matrix $Q$ we have that $||Qx|| = ||x||$. Thus in your computation we can write

$$ ||AQ||_2 \;\; =\;\; \sup_{||x||=1} ||AQx|| \;\; =\;\; \sup_{||Qx||=1} ||AQx|| \;\; =\;\; \sup_{||y||=1} ||Ay|| \;\; =\;\; ||A||_2. $$