The reverse order laws for {1, 2, 3}- and {1, 2, 4}-inverses of multiple matrix products

Abstract

In this article, we study the reverse order laws for {1, 2, 3}-and {1, 2, 4}-inverses of multiple matrix products by using the maximal and minimal ranks of the generalized Schur complements. The necessary and sufficient conditions for the inclusions A n f1, 2, 3gA nÀ1 f1, 2, 3g Á Á Á A 1 f1, 2, 3g ðA 1 A 2 Á Á Á A n Þf1, 2, 3g and A n f1, 2, 4gA nÀ1 f1, 2, 4g Á Á Á A 1 f1, 2, 4g ðA 1 A 2 Á Á Á A n Þf1, 2, 4g, are presented.

An invariance property related to the mixed-type reverse order laws

“…They have attracted considerable attention since the middle 1960s, [11] and recently many interesting results have been obtained, see, eg. [4,[12][13][14][15][16][17][18][19][20]. Let A ∈ C m×n and B ∈ C n× p , a solution to the reverse order law B{1}A{1} ⊆ (AB){1} is recalled in the following.…”

The forward order laws for {1,2,3}- and {1,2,4}-inverses of a three matrix products

“…Theory and computations of the reverse order laws for generalized inverses of matrix product are important subjects in many branches of applied science, such as non-linear control theory, matrix theory, matrix algebra, see [6,8,9,13]. One of the core problems in reverse order laws is to find the necessary and sufficient conditions for the reverse order laws for the generalized inverse of matrix product and it has attracted considerable attention, see [1,2,7,9,18,19].…”

“…With the same method, Wei [2,15], Wei and Guo [16] studied the equivalent conditions for the reverse order laws of {1}-inverses, {1, 2}-inverses, {1, 3}-inverses and {1, 4}-inverses of multiple matrix products. During the recent years, Zheng and Xiong [18,19] studied the reverse order laws for {1, 2, 3}-inverses and {1, 2, 4}-inverses of multiple products. For other interesting results on this subject see [1,2,8,9,14].…”

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].…”