The question is that:
For $n\times n$ matrices, could we define a norm,
(1)s.t. $$\forall A, B \in M_n(\mathbb{K}), ||A.B||=||B.A||$$
(2)or is there a norm s.t. $$\forall A\in M_n{(\mathbb{K})}, \forall P\in GL_n(\mathbb{K}), ||P^{-1}AP||=||A||$$
I think the 2 problems have the same idea and both are wrong, but I cannot find a conterexample for neither...
Could someone give a hint?
Thanks~