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Group inverse for the block matrices with an invertible subblock. (English) Zbl 1181.15003
Suppose that $M = \left(\matrix A & B \\ C & D \endmatrix \right)$ ($A$ is square) is a square block matrix with an invertible subblock over a skew field $K$. The authors presented some necessary and sufficient conditions for the existence as well as the expressions of the group inverse for $M$ under some conditions.

15A09Matrix inversion, generalized inverses
15B33Matrices over special rings (quaternions, finite fields, etc.)
Full Text: DOI
[1] S.L. Campbell, C.D. Meyer, Generalized inverses of linear transformations, Pitman, London, 1979, Dover, New York, 1991. · Zbl 0417.15002
[2] A. Ben-Israel, T.N.E. Greville, Generalized Inverses: theory and applications, second ed., Wiley, New York, 1974, Springer-Verlag, New York, 2003. · Zbl 0305.15001
[3] Campbell, S. L.: The Drazin inverse and systems of second order linear differential equations, Linear multilinear algebra 14, 195-198 (1983) · Zbl 0523.15007 · doi:10.1080/03081088308817556
[4] Wei, Y.; Diao, H.: On group inverse of singular Toeplitz matrices, Linear algebra appl. 399, 109-123 (2005) · Zbl 1072.15007 · doi:10.1016/j.laa.2004.08.021
[5] Kirkland, S. J.; Neumann, M.: On group inverses of M-matrices with uniform diagonal entries, Linear algebra appl. 296, 153-170 (1999) · Zbl 0933.15007 · doi:10.1016/S0024-3795(99)00127-5
[6] Heinig, G.: The group inverse of the transformation ϕ$(X)=$AX-XB, Linear algebra appl. 257, 321-342 (1997) · Zbl 0874.15004
[7] Chen, X.; Hartwig, R. E.: The group inverse of a triangular matrix, Linear algebra appl. 237/238, 97-108 (1996) · Zbl 0851.15005 · doi:10.1016/0024-3795(95)00561-7
[8] Hartwig, R. E.; Shoaf, J. M.: Group inverse and Drazin inverse of bidiagonal and triangular Toeplitz matrices, Aust. J. Math. 24, No. A, 10-34 (1977) · Zbl 0372.15003
[9] Meyer, C. D.; Rose, N. J.: The index and the Drazin inverse of block triangular matrices, SIAM J. Appl. math. 33, 1-7 (1977) · Zbl 0355.15009 · doi:10.1137/0133001
[10] Li, X.; Wei, Y.: A note on the representations for the Drazin inverse of $2\times 2$ block matrices, Linear algebra appl. 423, 332-338 (2007) · Zbl 1121.15008 · doi:10.1016/j.laa.2007.01.005
[11] Cvetković-Ilić, D. S.: A note on the representation for the Drazin inverse of $2\times 2$ block matrices, Linear algebra appl. 429, 242-248 (2008) · Zbl 1148.15001 · doi:10.1016/j.laa.2008.02.019
[12] Castro-González, N.; Dopazo, E.: Representations of the Drazin inverse of a class of block matrices, Linear algebra appl. 400, 253-269 (2005) · Zbl 1076.15007 · doi:10.1016/j.laa.2004.12.027
[13] Bu, C.; Zhao, J.; Zheng, J.: Group inverse for a class $2\times 2$ block matrices over skew fields, Appl. math. Comput. 204, 45-49 (2008) · Zbl 1159.15003 · doi:10.1016/j.amc.2008.05.145
[14] Bu, C.: On group inverses of block matrices over skew fields, J. math. 26, No. 1, 49-52 (2006) · Zbl 1101.15001
[15] Castro-González, N.; Dopazo, E.; Robles, J.: Formulas for the Drazin inverse of special block matrices, Appl. math. Comput. 174, 252-270 (2006) · Zbl 1097.15005 · doi:10.1016/j.amc.2005.03.027
[16] Cao, C.: Some results of group inverses for partitioned matrices over skew fields, J. natural sci. Heilongjiang univ. 18, No. 3, 5-7 (2001) · Zbl 1076.15506
[17] Cao, C.; Tang, X.: Representations of the group inverse of some $2\times 2$ block matrices, Int. math. Forum 31, 1511-1517 (2006) · Zbl 1119.15002
[18] Wei, Y.: Expression for the Drazin inverse of a $2\times 2$ block matrix, Linear multilinear algebra 45, 131-146 (1998)
[19] Hartwig, R.; Li, X.; Wei, Y.: Representations for the Drazin inverse of $2\times 2$ block matrix, SIAM J. Matrix anal. Appl. 27, 757-771 (2006) · Zbl 1100.15003 · doi:10.1137/040606685
[20] Rao, K. P. S. Bhaskara: The theory of generalized inverses over commutative rings, (2002) · Zbl 0992.15003
[21] Zhuang, W.: The guidance of matrix theory over skew fields, (2006)