Rank Function And Outer Inverses

Karantha, Manjunatha Prasad and Bhat, Nayan K and Mishra, Nupur Nandini (2018) Rank Function And Outer Inverses. Electronic Journal of Linear Algebra, 33. pp. 16-23. ISSN 1081-3810

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Abstract

For the class of matrices over a �eld, the notion of `rank of a matrix' as de�ned by `the dimension of subspace generated by columns of that matrix' is folklore and cannot be generalized to the class of matrices over an arbitrary commutative ring. The `determinantal rank' de�ned by the size of largest submatrix having nonzero determinant, which is same as the column rank of given matrix when the commutative ring under consideration is a �eld, was considered to be the best alternative for the `rank' in the class of matrices over a commutative ring. Even this determinantal rank and the McCoy rank are not so e�cient in describing several characteristics of matrices like in the case of discussing solvability of linear system. In the present article,the `rank{function' associated with the matrix as de�ned in [Solvability of linear equations and rank{function, K. Manjunatha Prasad, http://dx.doi.org/10.1080/00927879708825854] is discussed and the same is used to provide a necessary and su�cient condition for the existence of an outer inverse with speci�c column space and row space. Also, a rank condition is presented for the existence of Drazin inverse, as a special case of an outer inverse, and an iterative procedure to verify the same in terms of sum of principal minors of the given square matrix over a commutative ring is discussed

Item Type: Article
Uncontrolled Keywords: Rank-function; Generalized inverse; Outer inverse; Drazin inverse; Matrix over a commutative ring
Subjects: Departments at MU > Statistics
Depositing User: KMC Library
Date Deposited: 05 Oct 2019 05:08
Last Modified: 05 Oct 2019 05:08
URI: http://eprints.manipal.edu/id/eprint/154699

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