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A study of range equalities for mixed products of two matrices and their generalized inverses

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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 their generalized inverses. This article considers a fundamental problem in theory of generalized inverses of matrices on characterizing range equalities composed of matrices and their generalized inverses. The coverage includes providing some general principles of dealing with this kind of problems, establishing a group of necessary and sufficient conditions for the equality \(\mathrm{range}(D_1 - C_1A_1^{\dag }B_1) = \mathrm{range}(D_2 - C_2A_2^{\dag }B_2)\) to hold for the Moore–Penrose generalized inverses \(A_1^{\dag }\) and \(A_2^{\dag }\) through the skillful use of the matrix rank method and the block matrix representation method, establishing a variety of range equalities with extrusion properties for multiple matrix products composed of two matrices and their conjugate transposes and Moore–Penrose generalized inverses, and deriving a group of novel necessary and sufficient conditions for the reverse-order law to hold via the matrix range equality method.

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Acknowledgements

The author wishes to thank referees for their thorough and scholarly reports to the earlier versions of this article.

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  1. Shanghai Business School, Shanghai, 201400, China

    Yongge Tian

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  1. Yongge Tian
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Correspondence to Yongge Tian.

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Communicated by Margherita Porcelli.

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Tian, Y. A study of range equalities for mixed products of two matrices and their generalized inverses. Comp. Appl. Math. 41, 384 (2022). https://doi.org/10.1007/s40314-022-02084-x

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  • DOI: https://doi.org/10.1007/s40314-022-02084-x

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