Matrix norms using trace

Question

Show that $||A||^2_F = tr(A^TA)$ is a norm, but not an induced norm.

i know trace of a matrix is not a norm since norm of A=0 iff A=0. $A^TA$ is positive definite here but howd that help me? and what is f space here?

Answer 1

You can verify that $<A,B> = tr(A^T B) $ is an inner product from the axioms. Then, this is the norm induced by that inner product (clearly, $ <A,A> \geq 0$ when does it equal zero?, $<A,B> = tr(A^T B) = tr( (A^T B)^T ) = tr(B^T A) = <B,A>$, etc. ). If you have an inner product $<\cdot, \cdot>$, $\sqrt{<\cdot,\cdot>}$ is a norm.

An induced norm is one that satisfies $\lVert A \rVert = \sup_{ \lVert x \rVert = 1} \lVert A x \rVert_v$ for some vector norm $\lVert \cdot \rVert_v$. You want to show this is not the case for any vector norm. Note that if $A = I$, then the definition requires $\lVert A \rVert = 1$ (which is violated by frobenius norm).