Let H be a linear subspace of the space of Hermitian ntimesn matrices. Is there a good characterization of those H such that every AinH 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)=mathrmtr(AB)), or equivalently there exists a basis of mathbbCn such that all matrices in H have zero trace.
Even for k=2 I was not able to find any good characterization.
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