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 L\"owner 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.