Abstract : We show that the set of maximal lower bounds of two symmetric matrices with respect to Loewner order can be identified to the quotient set O(p,q)/(O(p)×O(q)). Here, (p,q)denotes the inertia of the difference of the two matrices, O(p) is the p-th orthogonal group, and O(p,q) is the indefinite orthogonal group arising from a quadratic form with inertia (p,q). We discuss the application of this result to the synthesis of ellipsoidal invariants of hybrid dynamical systems.
