Rank equalities related to the generalized inverse \(A^{(2)}_{T,S}\) with applications. (English) Zbl 1159.15006

The authors establish the rank equalities of some matrix expressions and certain block matrices related to the generalized inverse \(A_{T,S}^{(2)}\). They define the block independence in the generalized inverse \(A_{T,S}^{(2)}\) and derive necessary and sufficient conditions for two, three and four ordered matrices to be independent in \(A_{T,S}^{(2)}\), respectively. As special cases, they present the corresponding results on the weighted Moore-Penrose inverse and the Drazin inverse. See Y.-H. Liu and M.-S. Wei [Acta Math. Sin., Engl. Ser. 23, No. 4, 723–730 (2007; Zbl 1123.15003)] and Y. Wang [SIAM J. Matrix Anal. Appl. 19, No. 2, 407–415 (1998; Zbl 0926.15003)] as some related materials.


15A09 Theory of matrix inversion and generalized inverses
15A18 Eigenvalues, singular values, and eigenvectors
15A03 Vector spaces, linear dependence, rank, lineability
Full Text: DOI


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