Assume all the matrices I discuss about are $N \times N$. Consider any two hermitian matrices $A_1$ and $A_2$ which are indefinite. The question is, In general, for any $A_1$ and $A_2$ (both matrices are guaranteed to have at least one zero eigen value), does there exist any positive number $t$ such that equation \begin{align} (tA_1+(1−t)A_2)x=0 \end{align} has a non-zero vector $x$ as a solution.

( This is my first question in mathoverflow. I am not a mathematician, but from an engineering back ground. In my application, this kind of problem arises. My level of mathematical maturity is not enough to solve it. I hope some one here can.)