## Journal of Applied Mathematics

### New Representations of the Group Inverse of $2×2$ Block Matrices

#### Abstract

This paper presents a full rank factorization of a $2×2$ block matrix without any restriction concerning the group inverse. Applying this factorization, we obtain an explicit representation of the group inverse in terms of four individual blocks of the partitioned matrix without certain restriction. We also derive some important coincidence theorems, including the expressions of the group inverse with Banachiewicz-Schur forms.

#### Article information

Source
J. Appl. Math., Volume 2013 (2013), Article ID 247028, 10 pages.

Dates
First available in Project Euclid: 14 March 2014

https://projecteuclid.org/euclid.jam/1394808064

Digital Object Identifier
doi:10.1155/2013/247028

Mathematical Reviews number (MathSciNet)
MR3082046

Zentralblatt MATH identifier
1271.15001

#### Citation

Liu, Xiaoji; Yang, Qi; Jin, Hongwei. New Representations of the Group Inverse of $2×2$ Block Matrices. J. Appl. Math. 2013 (2013), Article ID 247028, 10 pages. doi:10.1155/2013/247028. https://projecteuclid.org/euclid.jam/1394808064

#### References

• F. J. Hall, “Generalized inverses of a bordered matrix of operators,” SIAM Journal on Applied Mathematics, vol. 29, pp. 152–163, 1975.
• F. J. Hall, “The Moore-Penrose inverse of particular bordered matrices,” Australian Mathematical Society Journal A, vol. 27, no. 4, pp. 467–478, 1979.
• F. J. Hall and R. E. Hartwig, “Further results on generalized inverses of partitioned matrices,” SIAM Journal on Applied Mathematics, vol. 30, no. 4, pp. 617–624, 1976.
• S. K. Mitra, “Properties of the fundamental bordered matrix used in linear estimation,” in Statistics and Probability, Essays in Honor of C.R. Rao, G. Kallianpur, Ed., pp. 504–509, North Holland, New York, NY, USA, 1982.
• J. K. Baksalary and G. P. H. Styan, “Generalized inverses of partitioned matrices in Banachiewicz-Schur form,” Linear Algebra and its Applications, vol. 354, pp. 41–47, 2002.
• F. Burns, D. Carlson, E. Haynsworth, and T. Markham, “Generalized inverse formulas using the Schur complement,” SIAM Journal on Applied Mathematics, vol. 26, pp. 254–259, 1974.
• D. S. Cvetković-Ilić, “A note on the representation for the Drazin inverse of $2\times 2$ block matrices,” Linear Algebra and its Applications, vol. 429, no. 1, pp. 242–248, 2008.
• G. Marsaglia and G. P. H. Styan, “Rank conditions for generalized inverses of partitioned matrices,” Sankhyā A, vol. 36, no. 4, pp. 437–442, 1974.
• J. Benítez and N. Thome, “The generalized Schur complement in group inverses and $(k+1)$-potent matrices,” Linear and Multilinear Algebra, vol. 54, no. 6, pp. 405–413, 2006.
• D. S. Cvetković-Ilić, J. Chen, and Z. Xu, “Explicit representations of the Drazin inverse of block matrix and modified matrix,” Linear and Multilinear Algebra, vol. 57, no. 4, pp. 355–364, 2009.
• J. M. Miao, “General expressions for the Moore-Penrose inverse of a $2\times 2$ block matrix,” Linear Algebra and its Applications, vol. 151, pp. 1–15, 1991.
• J. Chen, Z. Xu, and Y. Wei, “Representations for the Drazin inverse of the sum $P+Q+R+S$ and its applications,” Linear Algebra and its Applications, vol. 430, no. 1, pp. 438–454, 2009.
• S. L. Campbell, C. D. Meyer, Jr., and N. J. Rose, “Applications of the Drazin inverse to linear systems of differential equations with singular constant coefficients,” SIAM Journal on Applied Mathematics, vol. 31, no. 3, pp. 411–425, 1976.
• R. Hartwig, X. Li, and Y. Wei, “Representations for the Drazin inverse of a $2\times 2$ block matrix,” SIAM Journal on Matrix Analysis and Applications, vol. 27, no. 3, pp. 757–771, 2005.
• A. da Silva Soares and G. Latouche, “The group inverse of finite homogeneous QBD processes,” Stochastic Models, vol. 18, no. 1, pp. 159–171, 2002.
• Y. Wei and H. Diao, “On group inverse of singular Toeplitz matrices,” Linear Algebra and its Applications, vol. 399, pp. 109–123, 2005.
• Y. Wei, “On the perturbation of the group inverse and oblique projection,” Applied Mathematics and Computation, vol. 98, no. 1, pp. 29–42, 1999.
• J. Benítez, X. Liu, and T. Zhu, “Additive results for the group inverse in an algebra with applications to block operators,” Linear and Multilinear Algebra, vol. 59, no. 3, pp. 279–289, 2011.
• C. Bu, M. Li, K. Zhang, and L. Zheng, “Group inverse for the block matrices with an invertible subblock,” Applied Mathematics and Computation, vol. 215, no. 1, pp. 132–139, 2009.
• C. Bu, K. Zhang, and J. Zhao, “Some results on the group inverse of the block matrix with a sub-block of linear combination or product combination of matrices over skew fields,” Linear and Multilinear Algebra, vol. 58, no. 7-8, pp. 957–966, 2010.
• C. Bu, J. Zhao, and K. Zhang, “Some results on group inverses of block matrices over skew fields,” Electronic Journal of Linear Algebra, vol. 18, pp. 117–125, 2009.
• C. Bu, J. Zhao, and J. Zheng, “Group inverse for a class $2\times 2$ block matrices over skew fields,” Applied Mathematics and Computation, vol. 204, no. 1, pp. 45–49, 2008.
• C. G. Cao, “Some results of group inverses for partitioned matrices over skew fields,” Journal of Natural Science of Heilongjiang University, vol. 18, no. 3, pp. 5–7, 2001 (Chinese).
• X. Chen and R. E. Hartwig, “The group inverse of a triangular matrix,” Linear Algebra and its Applications, vol. 237/238, pp. 97–108, 1996.
• C. Cao and J. Li, “Group inverses for matrices over a Bezout domain,” Electronic Journal of Linear Algebra, vol. 18, pp. 600–612, 2009.
• C. Cao and J. Li, “A note on the group inverse of some $2\times 2$ block matrices over skew fields,” Applied Mathematics and Computation, vol. 217, no. 24, pp. 10271–10277, 2011.
• C. Cao and X. Tang, “Representations of the group inverse of some $2\times 2$ block matrices,” International Mathematical Forum, vol. 31, pp. 1511–1517, 2006.
• M. Catral, D. D. Olesky, and P. van den Driessche, “Graphical description of group inverses of certain bipartite matrices,” Linear Algebra and its Applications, vol. 432, no. 1, pp. 36–52, 2010.
• P. Patrício and R. E. Hartwig, “The $(2,2,0)$ group inverse problem,” Applied Mathematics and Computation, vol. 217, no. 2, pp. 516–520, 2010.
• J. Zhou, C. Bu, and Y. Wei, “Group inverse for block matrices and some related sign analysis,” Linear and Multilinear Algebra, vol. 60, no. 6, pp. 669–681, 2012.
• Z. Yan, “New representations of the Moore-Penrose inverse of $2\times 2$ block matrices,” Linear Algebra and its Applications, 2013.