Maximal lower bounds in the Löwner order

Abstract

We show that the set of maximal lower bounds of two symmetric matrices with respect to the Löwner order can be identified to the quotient set $O(p,q)/(O(p)\times O(q))$. 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 also show that a similar result holds for positive semidefinite maximal lower bounds with maximal rank of two positive semidefinite matrices. We exhibit a correspondence between the maximal lower bounds $C$ of two matrices $A,B$ and certain pairs of subspaces, describing the directions on which the quadratic form associated with $C$ is tangent to the one associated with $A$ or $B$. The present results refines a theorem from Kadison that characterizes the existence of the infimum of two symmetric matrices and a theorem from Moreland, Gudder and Ando on the existence of the positive semidefinite infimum of two positive semidefinite matrices.