References for bounds on dimension of particular matrix spaces?

I'm looking for references and well-known results about bounds on dimension of particular matrix spaces.

For instance, the first result that came up to my mind was a Flander's theorem which explains that if we consider $F$ a vector subspace of $\mathcal{M}_n(\mathbb{K})$ (where $\mathbb{K}$ is a field) which contains only matrices of rank at most $r\le n$ then we obtain that $\dim(F) \le r\times n$.

Also we have a well-known theorem by Gerstenhaber about the space of nilpotent matrix. Is says that if $V$ is a vector subspace of $\mathcal{M}_n(\mathbb{K})$ which contains nilpotent matrices then the maximum of $\dim(V)$ is $\dfrac{n(n-1)}{2}$.

Quite recently in 2009, there was this result proven by two crazy mathematicians : if $\mathcal A$ is a nontrivial sub-algebra of $\mathcal{M}_n(\mathbb{K})$ with nontrivial centralizer then

$$\dim (\mathcal A) + \dim(\mathcal{C}(\mathcal A)) \le (n -1)^2 + 3$$

In some sense, maybe we can also consider the Sylvester's law of inertia. Moreover, we can add the theory of the centralizer of a matrix which gives results about dimensions.

Apparently there are results found by Toeplitz for these kind of questions. If someone has other references or know other results about these kind of statements, it will be perfect to share.

Thanks in advance !

For now, here are interesting references I found:

• Linear spaces of Toeplitz and nilpotent matrices (1993)

• http://dsp.prod.free.fr/recherche.html+ RMS 118-1 + RMS 118-4 + RMS 121-1