How to solve three fundamental linear matrix inequalities in the Löwner partial ordering

Abstract

This paper shows how to solve analytically the three fundamental linear matrix inequalities $$AXB\succcurlyeq C(\succ C),AX{A}^{\ast }\succcurlyeq B(\succ B),AX+(AX{)}^{\ast }\succcurlyeq B(\succ B)$$ in the Löwner partial ordering by using ranks, inertias and generalized inverses of matrices.

37 pages