Maximal lower bounds in the L\"owner order

TL;DR

This paper characterizes the set of maximal lower bounds of two symmetric matrices under the L"owner order using quotient sets of orthogonal groups, extending classical theorems on matrix infima.

Contribution

It provides a geometric and group-theoretic description of maximal lower bounds, refining existing theorems on matrix infima and positive semidefinite bounds.

Findings

Set of maximal lower bounds identified with quotient of indefinite orthogonal group

Correspondence between bounds and pairs of subspaces with tangent quadratic forms

Refinement of Kadison's and Moreland-Gudder-Ando's theorems on matrix infima

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.