Two groups of mixed reverse order laws for generalized inverses of two and three matrix products

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Two groups of mixed reverse order laws for generalized inverses of two and three matrix products Yongge Tian1 Received: 9 November 2019 / Revised: 29 April 2020 / Accepted: 20 May 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020

Abstract Generalized inverses of a matrix product can be written as certain matrix expressions that are composed by the given matrices and their generalized inverses, and a challenging task in this respect is to establish various reasonable reverse order laws for generalized inverses of matrix products. In this paper, we present two groups of known and new mixed reverse order laws for the Moore–Penrose inverses of products of two and three matrices through various conventional matrix operations. We also establish four groups of matrix set inclusions that are composed by {1}- and {1, 2}-generalized inverses of A, B, C, and their products AB and ABC. Keywords Matrix product · Generalized inverse · Reverse order law · Set inclusion Mathematics Subject Classification 15A09 · 15A24 · 47A05

1 Introduction Throughout this article, let Cm×n denote the set of all m × n complex matrices, and let A∗ and Im denote the conjugate transpose and the identity matrix of order m, respectively. The Moore–Penrose inverse of A ∈ Cm×n , denoted by A† , is the unique matrix X ∈ Cn×m satisfying the four Penrose equations: (1) AX A = A, (2) X AX = X , (3) (AX )∗ = AX , (4) (X A)∗ = X A.


A matrix X is called a {i, . . . , j}-generalized inverse of A, denoted by A(i,..., j) , if it satisfies the ith,. . . , jth equations in (1.1). The collection of all {i, . . . , j}-generalized inverses of A is denoted by {A(i,..., j) }. There are all 15 types of {i, . . . , j}-generalized inverses of A, but A† , A(1,3,4) , A(1,2,4) , A(1,2,3) , A(1,4) , A(1,3) , A(1,2) , and A(1) are usually called the eight

Communicated by Jinyun Yuan.

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Yongge Tian [email protected] College of Business and Economics, Shanghai Business School, Shanghai, China 0123456789().: V,-vol



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Y. Tian

commonly used types of generalized inverses of A in the literature, see, e.g., Ben-Israel and Greville (2003); Campbell and Meyer (2009). As is known to all, formulations and characterizations of qualities and identities are undoubtedly one of the central tasks over various algebraic systems. In the theory of generalized inverses of matrices, such a task is specified as derivations of various equalities and identities that involve matrices and their generalized inverses. As usual, matrix equalities that are composed by generalized inverses can symbolically be written in the following general form:     (i ,..., j1 ) (i ,..., j2 ) (i ,..., jk ) (s ,...,t ) (s ,...,t ) (s ,...,t ) f A1 1 = g B1 1 1 , B2 2 2 , . . . , Bl l l , (1.2) , A2 2 , . . . , Ak k where f (·) and g(·) denote certain conventional algebraic operations of matrices. Recall that for any three nonsingular matrices A, B, and C of the same size, the matrix identities (AB)−1 = B −1 A−1 and (ABC)−1 = C −1 B −1 A−1 always hol