Let $A,B\in \mathbb{R}^{n\times n}$ be two symmetric matrices, then when dose the following system:
$x^T A x=0$ & $x^T Bx=0$,
has nonzero solutions?
I know there are some sufficient conditions, e.g, Finsler's Lemma, guarantee the existence, or disprove the existence of such solutions.
But is there any work gives a sufficient and necessary condition? Or at least, is there any new results about the above problem? I have tried to search on Google Scholar, but it seems that there are too few papers concerning this problem. Could anybody please give some suggestions or recommend some readings?