Orthogonal matrix norm

This holds for any norm induced by an inner product. This follows from $$\|QA\|=\sqrt{(QA,QA)}=\sqrt{(Q^TQA,A)}=\sqrt{(A,A)} = \|A\|$$

With $Q$ an orthonormal matrix, i.e., $Q^{-1}=Q^T$.

The operator norm $$ \|A\|=\max\{\|Ax\|_2:\ \|x\|=1\}, $$ where $\|\cdot\|_2$ is the Euclidean norm, also satisfies those two equalities. They follow easily from the fact that $\|y\|_2^2=y^Ty$, so $$\|Hx\|_2^2=(Hx)^THx=x^TH^THx=x^Tx=\|x\|_2^2.$$

The frobenius norm, $||A||_F^2 = tr(A A^T)$ satisfies this property. You can see this by noting $|| H A||_F^2 = tr( HA (HA)^T) = tr(H A A^T H^T) = tr(H^T H A A^T) = tr( I A A^T ) = tr(A A^T) = ||A||_F^2$.