Can Spectral Radius be a Norm on $M_{n\times n}(\mathbb{C})$?

Question

Let $n > 1$ be an integer. For each $A \in M_{n\times n}(\mathbb{C})$, let $||A || = \rho(A)$, the spectral radius of $A$. Does this turn $M_{n×n}(\mathbb{C})$ into a normed space?

I think the answer must be no and here is my counterexample, but I'm not sure about it.

$$B=\begin{bmatrix} 0&1\\0&0 \end{bmatrix}$$ $Spec(B)=\{0\}$, however $B\neq 0$.

Answer 1

That is correct. Since it is possible to have $\rho(A) = 0$ with $A \ne 0$, this is not a norm.

Answer 2

Moreover, in the OP example we have $\rho(B) = \rho(B^t) = 0$, while $\rho(B+B^t) = 1$, so $\rho$ is not subadditive, hence not even a seminorm.