Matrix Cubes Parameterized by Eigenvalues

Abstract

An elimination problem in semidefinite programming is solved by means of tensor algebra. It concerns families of matrix cube problems whose constraints are the minimum and maximum eigenvalue functions on an affine space of symmetric matrices. A linear matrix inequality (LMI) representation is given for the convex set of all feasible instances, and its boundary is studied from the perspective of algebraic geometry. This generalizes the known LMI representations of k-ellipses and k-ellipsoids.