A variety of matrices (or algebraic set) $V$ is a set of matrices over a field $F$ which are the common zeros of a set of polynomials (on their entries). Some varieties contain linear subspaces of matrices, and others do not. Examples:
- Every linear subspace is a variety.
- If $0\not\in V$ then there is no linear subspace inside $V$.
- If $\det(A)=0$ then $\det(\lambda A)=0$ for all $\lambda\in F$.
- If $V$ is the variety of $\det$ on $\text{Mat}_n(F)$ and $L$ denotes the subspace of all matrices whose first row is full of zeros, then $L\subseteq V$ and $\dim_F L=n^2-n$.
Given a variety $V$ inside $\text{Mat}_{m\times n}(F)$, define $$\text{ldim}(V):=\max\{\dim L \ | \ L\leq\text{Mat}_{m\times n}(F), L\subseteq V\}$$
and ldim$(V)=-1$ if there is no such $L$ (iff $0\not\in V$).
My questions are the following:
1) Given $V$, what tools do we have to find ldim$(V)$?
2) What do we know when $V$ are the determinantal varieties (those of matrices with rank no greater than a fixed $k$)?
3) Given $\text{Mat}_{m\times n}(F)$, what is the maximum ldim possible for varieties which are not subspaces themselves, and for which varieties is it attained? For square matrices, is it $n^2-n$ as in the third example?