Solutions of the matrix inequalities in the minus partial ordering and L?wner partial ordering

摘要

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 L?wner 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 L?wner 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 L?wner partial orderings, respectively, and show that these two types of shorted matrices in fact are the same.