# Existence condition of Real Eigenvalues for Non-Symmetric Real Matrix

I have to solve this eigenproblem:

$$Ax=\lambda Bx$$

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?

• Sorry but I don’t understand the set up for that problem, A and B are scalar multiple? – gimusi Jul 27 '18 at 8:33
• Problem should be clear now – iacopo Jul 27 '18 at 8:41