Is it correct to say that matrix norm that are unitarily invariant norm has this property $|||U||| = |||I_n|||$, where $U \in M_n$ is some unitary/permutation matrix, and $I_n$ is an identity matrix?
The definition of unitarily invariant norm is: $|||UAV|||=|||A|||$, where $U$ and $V$ are unitary matrices.
Thank you
p.s.: I get that $|||I|||$ is not necessarily equal to $1$ (unless it's an induced norm).