I happened to notice that there is concept "stable rank" that people used a lot in matrix computation theories, such as the work of Rudelson & Vershynin (2005). It is defined to be the ratio between squared Frobenius norm and the squared spectral norm of a matrix. $$r(M) = ||M||_F^2/||M||_2^2$$
I was wondering if there is any literature or reference that discuss some basic properties of this stable rank. For example, do we know $$r(M_1+M_2) \le r(M_1) + r(M_2)$$ which is expected to be true since the stable rank is expected to be more stable than rank?
Thanks.