Orthogonal matrix norm If $H$ is an orthogonal matrix, then $||H||=1$ and $||HA||=||A||, \forall A$-matrix (such that we can writ $H \cdot A$). What norm is this about?

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., $Q^{-1}=Q^T$.