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Generalized inverses over integral domains. II: Group inverses and Drazin inverses. (English) Zbl 0722.15007

This is a continuation of an earlier paper by the authors [ibid. 140, 181–196 (1990; Zbl 0712.15004)] on generalized inverses over integral domains. The main results consist of necessary and sufficient conditions for the existence of a group inverse, a new formula for a group inverse when it exists, and necessary and sufficient conditions for the existence of a Drazin inverse. The authors show that a square matrix \(A\) of rank \(r\) over an integral domain \({\mathcal R}\) has a group inverse if and only if the sum of all \(r\times r\) principal minors of \(A\) is an invertible element of \({\mathcal R}\). They also show that when it exists, the group inverse of \(A\) is a polynomial in \(A\) with coefficients from \({\mathcal R}\).

MSC:

15A09 Theory of matrix inversion and generalized inverses
15B33 Matrices over special rings (quaternions, finite fields, etc.)

Citations:

Zbl 0712.15004
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References:

[1] R.B. Bapat, K.P.S. Bhaskara Rao, and K. Manjunatha Prasad, Generalized inverses over integral domains, Linear Algebra Appl.; R.B. Bapat, K.P.S. Bhaskara Rao, and K. Manjunatha Prasad, Generalized inverses over integral domains, Linear Algebra Appl. · Zbl 0712.15004
[2] Ben-Israel, A.; Greville, T. N.E., Generalized Inverses and Applications (1974), Wiley · Zbl 0305.15001
[3] Bhaskara Rao, K. P.S., On generalized inverses of matrices over integral domains, Linear Algebra Appl., 49, 179-189 (1983) · Zbl 0505.15002
[4] Rao, C. R.; Mitra, S. K., Generalized Inverse of Matrices and Applications (1971), Wiley · Zbl 0236.15004
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