Let $H$ be a linear subspace of the space of Hermitian $n\times n$ matrices. Is there a good characterization of those $H$ such that every $A\in H$ has at least $k$ positive and $k$ negative eigenvalues?
For $k=1$ a nice characterization is the following: there is a positive definite matrix $B$ orthogonal to $H$ (w.r.t. the scalar product $(A,B)=\mathrm{tr}(AB)$), or equivalently there exists a basis of $\mathbb C^n$ such that all matrices in $H$ have zero trace.
Even for $k=2$ I was not able to find any good characterization.
Best Answer
In an 2n-dimensional space the space of block matrices of the form
0 A*
A 0
have n positive and n negative eigenvalues.
They are plus or minus the singular values of A. (Meaning eigenvalues of |A|=(A*A)^(1/2)).
(This fact is in Bhatia's matrix analysis book.)