  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.
