Equivalence analysis of different reverse order laws for generalized inverses of a matrix product

Yongge Tian
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Yongge Tian: CBE, Shanghai Business School

Indian Journal of Pure and Applied Mathematics, 2022, vol. 53, issue 4, 939-947

Abstract: Abstract Generalized inverses of matrices are extensions of ordinary inverses of nonsingular matrices to singular matrices, and people can construct various matrix expressions and equalities that involve matrices and generalized inverses. A famous reverse order law (ROL) for the Moore–Penrose generalized inverses of matrix products is $$(AB)^{\dag } = B^{\dag }A^{\dag }$$ ( A B ) † = B † A † , which is an extension of the reverse order law $$(AB)^{-1} = B^{-1}A^{-1}$$ ( A B ) - 1 = B - 1 A - 1 for the ordinary inverse of the product of two nonsingular matrices of the same size. The aim of this article is to formulate a number of reasonable matrix equalities that are composed of the products of AB, $$(AB)^{\dag }$$ ( A B ) † , and $$B^{\dag }A^{\dag }$$ B † A † , and to present many new facts that are equivalent to $$(AB)^{\dag } = B^{\dag }A^{\dag }$$ ( A B ) † = B † A † and its variations by means of the matrix rank method and the matrix range method.

Keywords: matrix product; Moore–Penrose generalized inverse; reverse order law; orthogonal projector; rank; range; 15A09; 15A24 (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s13226-021-00200-x

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