a norm is symmetric if and only if it is unitarily invariant how can I prove this :
A norm on $\mathcal{M}_n(\mathbb{C})$ is symmetric if and only if it is
unitarily invariant  ?
My attempt
I know that a symmetric norm is a norm which verifies : $$N(ABC)\leq N(B)N(A)N(C) $$ and I know that a norm is unitarily invariant if it verifies for unitary matrix $Q ,Q'$ $ N(QAQ') =N(A) $  but I don't know how to prove the statement.


This is from R. Bhatia book page 94.
The definition you wrote of the symmetric norm is wrong. This is the true definition and that’s the proof
