Conditions for Eigenvalue Configurations of Two Real Symmetric Matrices

In this extended abstract, we describe recent progress on finding conditions for eigenvalue configurations of two real symmetric matrices. To illustrate the problem, consider a simple example. Let F and G be 2 × 2 real symmetric matrices. Since the matrices are symmetric, their eigenvalues are all real. Thus we may consider the configuration (relative locations of) of the two eigenvalues of F and the two eigenvalues of G on the real line. The problem is, given a certain configuration of those eigenvalues, to find a simple condition on the entries of F and G so that the eigenvalues satisfy the given configuration. For a more precise statement of the problem, see Section 1.