Rank as norm on matrix

Question

Could we consider matrix rank $r$ a norm?

Is other norm similar to rank $r$ possible to associate with a finite matrix?

(We denote trivial valuation $|\alpha|=1,\quad\forall\alpha\in\Bbb F^*$ where $\Bbb F$ is field/division algebra).

It seems to satisfy norm axioms:

$1$ $\mathsf{rank}(M)=0\iff M=0$.

$2$ $\mathsf{rank}(\alpha M)=|\alpha|\mathsf{rank}(M)=\mathsf{rank}(M)$.

$3$ $\mathsf{rank}(M+N)\leq \mathsf{rank}(M)+\mathsf{rank}(N)$.

Answer 1

Yes, this is possible.

In fact, the theory of of rank error correcting codes makes use of this observation: The rank provides an absolute value of the set of all $m\times n$-matrices. Hence $d(A,B) = \operatorname{rk}(A - B)$ is a metric.