The reverse order laws for {1,2,3} - and {1,2,4} -inverses of a two-matrix product

Abstract

In this work we study the reverse order laws for {1, 2, 3}-and {1, 2, 4}-inverses of a product of two matrices by using the expressions for maximal and minimal ranks of the generalized Schur complement. The necessary and sufficient conditions for B{1, 2, 3}A{1, 2, 3} ⊆ (AB){1, 2, 3} and B{1, 2, 4}A{1, 2, 4} ⊆ (AB){1, 2, 4} are presented.

The mixed-type reverse order laws for weighted generalized inverses of a triple matrix product

“…Then (a) There exist (AB) (1,4) and (BC) (1,4) such that (BC) (1,4) B(AB) (1,4) Theorem 3. 5 Let A ∈ C m×n , B ∈ C n×p , C ∈ C p×q ; M ∈ C m×m , N ∈ C n×n , P ∈ C p×p and Q ∈ C q×q be four positive definite Hermitian matrices, let Z = ABC.…”

“…There always exist (AB) (1,4) and (BC) (1,4) such that (BC) (1,4) B(AB) (1,4) In this section, we investigate the two mixed-type reverse order laws in (1.7) and (1.8).…”

“…The reverse order laws for the generalized inverses of the multiple matrix products yields a class of interesting problems that are fundamental in the theory of generalized inverses of matrices and statistics. They have attracted considerable attention since the middle 1960s, and many interesting results have been obtained; see [4][5][6][7][8][9][12][13][14][15][16].…”

Some results on partial ordering and reverse order law of elements of C∗-algebras

“…An algebraic proof of the reverse order law for the Moore-Penrose inverse (in a ring with involution) is given in [17]. The interested reader can also consult [7,19].…”

Mixed-Type Reverse-Order Laws for the Generalized Inverses of an Operator Product

“…The reverse-order laws for the generalized inverses of an operator product yield a class of interesting problems which are fundamental in the theory of generalized inverses of operators. They have attracted considerable attention since the middle 1960s, and many interesting results have been obtained; see [4][5][6]8,9,11,[13][14][15][16][18][19][20][21][22][23][24][25][26].…”