I have to solve this eigenproblem:
Where $A$ and $B$ are generic submatrices (not principal) obtained from symmetric and real matrices $A_s$ and $B_s$ so it is obvious that in general $A$ and $B$ are non symmetric but real matrix. With $\lambda$ eigenvalue and $x$ the corresponding eigenvector. I know that for principal submatrices hold the Interlacing theorem since $A_s$ and $B_s$ are symmetric and real.
But, which conditions hold about the existence of real eigenvalues for the other submatrices?