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We show that the set of maximal lower bounds of two symmetric matrices withrespect 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 differenceof the two matrices, $O(p)$ is the $p$-th orthogonal group, and $O(p,q)$ is theindefinite orthogonal group arising from a quadratic form with inertia $(p,q)$.We also show that a similar result holds for positive semidefinite maximallower bounds with maximal rank of two positive semidefinite matrices. Weexhibit a correspondence between the maximal lower bounds $C$ of two matrices$A,B$ and certain pairs of subspaces, describing the directions on which thequadratic form associated with $C$ is tangent to the one associated with $A$ or$B$. The present results refines a theorem from Kadison that characterizes theexistence of the infimum of two symmetric matrices and a theorem from Moreland,Gudder and Ando on the existence of the positive semidefinite infimum of twopositive semidefinite matrices.

Maximal lower bounds in the L\"owner order