Nonlinear Kalman varieties

A common problem in quantum chemistry concerns finding eigenvectors of the Hamiltonian H. This matrix is usually very large and has to be approximated by smaller matrices when doing explicit computations. The eigenvectors of the Hamiltonian are constrained, meaning that quantum chemists know explicit equations that are fulfilled by some eigenvector of H. When approximating the matrix H arbitrarily, this additional information is usually lost due to the imprecise nature of the approximation of H. There are two ways to handle this issue: Either we relax the condition of "being an eigenvector the approximation" in trade of the condition that an "almost eigenvector" satisfies the constraints, or we choose the approximation of H in a clever way such that it has an eigenvector that exactly satisfies the constraints. In this talk we focus on the second approach which leads us to study nonlinear Kalman varieties. These are varieties of matrices which admit an eigenvector that satisfies certain nonlinear constraints. Inspired by the linear case, which was studied by Ottaviani and Sturmfels, we discuss the basic questions that an algebraic geometer might ask about these varieties, like dimension, degree and singularities. We also present a construction that produces explicit equations for Kalman varieties based on the minors of a large matrix with polynomial entries.