How to solve three fundamental linear matrix inequalities in the L\"owner partial ordering

TL;DR

This paper provides analytical solutions to three fundamental linear matrix inequalities in the L"owner partial ordering using ranks, inertias, and generalized inverses of matrices.

Contribution

It introduces a unified approach to solve key matrix inequalities analytically, leveraging matrix ranks, inertias, and generalized inverses.

Findings

Explicit analytical solutions for the inequalities.

Conditions for the inequalities to hold.

Method applicable to various matrix problems.

Abstract

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