Equalities and inequalities for ranks of products of generalized inverses of two matrices and their applications

Abstract

Abstract A complex matrix X is called an { i , … , j } $\{i,\ldots, j\}$ -inverse of the complex matrix A, denoted by A ( i , … , j ) $A^{(i,\ldots, j)}$ , if it satisfies the ith, …, jth equations of the four matrix equations (i) A X A = A $AXA = A$ , (ii) X A X = X $XAX=X$ , (iii) ( A X ) ∗ = A X $(AX)^{*} = AX$ , (iv) ( X A ) ∗ = X A $(XA)^{*} = XA$ . The eight frequently used generalized inverses of A are A † $A^{\dagger}$ , A ( 1 , 3 , 4 ) $A^{(1,3,4)}$ , A ( 1 , 2 , 4 ) $A^{(1,2,4)}$ , A ( 1 , 2 , 3 ) $A^{(1,2,3)}$ , A ( 1 , 4 ) $A^{(1,4)}$ , A ( 1 , 3 ) $A^{(1,3)}$ , A ( 1 , 2 ) $A^{(1,2)}$ , and A ( 1 ) $A^{(1)}$ . The { i , … , j } $\{i,\ldots, j\}$ -inverse of a matrix is not necessarily unique and their general expressions can be written as certain linear or quadratic matrix-valued functions that involve one or more variable matrices. Let A and B be two complex matrices such that the product AB is defined, and let A ( i , … , j ) $A^{(i,\ldots ,j)}$ and B ( i , … , j ) $B^{(i,\ldots,j)}$ be the { i , … , j } $\{i,\ldots, j\}$ -inverses of A and B, respectively. A prominent problem in the theory of generalized inverses is concerned with the reverse-order law ( A B ) ( i , … , j ) = B ( i , … , j ) A ( i , … , j ) $(AB)^{(i,\ldots,j)} = B^{(i,\ldots,j)}A^{(i,\ldots,j)}$ . Because the reverse-order products B ( i , … , j ) A ( i , … , j ) $B^{(i,\ldots,j)}A^{(i,\ldots,j)}$ are usually not unique and can be written as linear or nonlinear matrix-valued functions with one or more variable matrices, the reverse-order laws are in fact linear or nonlinear matrix equations with multiple variable matrices. Thus, it is a tremendous and challenging work to establish necessary and sufficient conditions for all these reverse-order laws to hold. In order to make sufficient preparations in characterizing the reverse-order laws, we study in this paper the algebraic performances of the products B ( i , … , j ) A ( i , … , j ) $B^{(i,\ldots,j)}A^{(i,\ldots,j)}$ . We first establish 126 analytical formulas for calculating the global maximum and minimum ranks of B ( i , … , j ) A ( i , … , j ) $B^{(i,\ldots,j)}A^{(i,\ldots,j)}$ for the eight frequently used { i , … , j } $\{i,\ldots, j\}$ -inverses of matrices A ( i , … , j ) $A^{(i,\ldots,j)}$ and B ( i , … , j ) $B^{(i,\ldots,j)}$ , and then use the rank formulas to characterize a variety of algebraic properties of these matrix products.

Keywords

• matrix product
• generalized inverse
• reverse-order law
• rank
• equality
• inequality