Solutions of the matrix inequalities in the minus partial ordering and Löwner partial ordering
Yongge Tian · Mathematical Inequalities & Applications · 2013
Two matrices A and B of the same size are said to satisfy the minus partial ordering, denoted by B -A , iff the rank subtractivity equality rank( A -B ) = rank(A)rank(B) holds; two complex Hermitian matrices A and B of the same size are said to satisfy the Lwner partial ordering, denoted by B L A , iff the difference A -B is nonnegative definite. In this note, we establish general solution of the inequality BXB * -A induced from the minus partial ordering, and general solution of the inequality BXB * L A induced from the Lwner partial ordering, respectively, where () * denotes the conjugate transpose of a complex matrix. As consequences, we give closed-form expressions for the shorted matrices of A relative to the range of B in the minus and Lwner partial orderings, respectively, and show that these two types of shorted matrices in fact are the same.