Conditions for eigenvalue configurations of two real symmetric matrices

Abstract: For two real symmetric matrices, their eigenvalue configuration is the relative arrangement of their eigenvalues on the real line. We consider the following problem: given an eigenvalue configuration, find a condition on the entries of two real symmetric matrices such that they have the given eigenvalue configuration. The problem amounts to finding a finite set of polynomials in the entries of the two matrices (which we call the configuration discriminant), and a way to express the eigenvalue configuration condition as a boolean expression of inequalities on the discriminant polynomials (which we call the configuration-from-sign transform). In this paper, we consider the problem under a mild condition that the two matrices do not share any eigenvalues. We approach the problem by reducing it to several classical real root counting problems for certain related polynomials.