Properties of Matrix Norms

• lemma: (one-sided) rotationally invariant matrix norm is orthogonally invariant iff reflectionally invariant
• proposition: for matrix norms induced by vector norms
  • left rotationally invariant iff codomain uses Euclidean norm
  • right rotationally invariant iff domain uses Euclidean norm
• proposition: for matrix norms induced by inner product
  • one-sided rotational invariance (any) implies scalar multiple of Frobenius norm implies unitarily invariant
• theorem (Von Neumann): matrix norm unitarily invariant iff norm equals some symmetric gauge function of singular values
  • special case: all Schatten p-norms