Consider two real symmetric matrices F and G. By eigenvalue configuration of F and G, we mean the relative locations of the eigenvalues of F and the eigenvalues of G on the real line.

We tackle the following problem:  given an eigenvalue configuration, find a simple condition on the entries of F and G so that the eigenvalues satisfy the given configuration.

Our motivation for tackling the eigenvalue configuration problem comes from two sources. First, it is a natural generalization of the celebrated Descartes' rule of sign. Second, many non-trivial problems in science and engineering can often be reduced to the problem.

We will describe an approach (or two if time allows) to the eigenvalue configuration problem.

This is a joint work with Daniel Profili and J. Rafael Sendra.