Let $K$ be a field, if necessary algebraic closed or of characteristic zero. Let $k$ be a positive integer. I am interested in linear subspaces $M \subseteq \textrm{Sym}\_n(K)$, where $\textrm{Sym}\_n(K)$ is the vector space of all symmetric $n \times n$ matrices with entries in $K$, such that every matrix in $M$ has at most rank $k$.


Are there any results on this problem? Is there a characterization of such subspaces?


If we do not restrict to symmetric matrices, but consider vector spaces of $m \times n$ matrices of rank at most $k$, this question was solved in the 80's for $k\leq 3$ by Atkinson and independently by Eisenbud and Harris. But the case $k>4$ is yet unknown, as far as I know. I think that if we restrict to symmetric matrices, this should be far less complicated, so perhaps there are results for $k>4$.


Thanks in advance!



Vector Spaces of Symmetric Matrices of Low Rank
There are three references you may want to check out: 


* R. Meshulam, On two extremal matrix problems, Linear Algebra Appl. 114/115 (1989), 261--271
* R. Loewy, N. Radwan, Spaces of symmetric matrices of bounded rank, Linear Algebra Appl. 197-198 (1994), 189--215
* R. Loewy, Large spaces of symmetric matrices of bounded rank are decomposable. Lin. Multlin. Alg. 48 (2001), 355--382.


In all those papers, the authors focus on subspaces of symmetric matrices with rank bounded above by $r$, and more precisely on the maximal dimension for these subspaces and on the structure of the spaces whose dimension is close to the maximal one. 


