Matrix group monotonicity using a dominance notion

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2015, 5(3): 267-274. doi: 10.3934/naco.2015.5.267

Matrix group monotonicity using a dominance notion

Debasisha Mishra ^1,

1.	Department of Mathematics, National Institute of Technology Raipur, Raipur- 492010, India

Received March 2013 Revised March 2015 Published August 2015

Abstract
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A dominance rule for group invertible matrices using proper splitting is proposed, and used this notion to show that a matrix is group monotone. Then some possible applications are discussed.

Keywords: group inverse, Drazin inverse, nonnegativity., Index-proper splittings.

Mathematics Subject Classification: Primary: 15A0.

Citation: Debasisha Mishra. Matrix group monotonicity using a dominance notion. Numerical Algebra, Control and Optimization, 2015, 5 (3) : 267-274. doi: 10.3934/naco.2015.5.267

References:

[1]	A. Ben-Israel and T. N. E. Greville, Generalized Inverses. Theory and Applications, Springer-Verlag, New York, 2003.
[2]	A. Berman and R. J. Plemmons, Cones and iterative methods for best square least squares solutions of linear systems, SIAM J. Numer. Anal., 11 (1974), 145-154.
[3]	A. Berman and R. J. Plemmons, Monotonicity and the generalized inverse, SIAM J. Appl. Math., 22 (1972), 155-161.
[4]	A. Berman and R. J. Plemmons, Matrix group monotonicity, Proceedings of the American Mathematical Society, 46 (1974), 355-359.
[5]	A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, SIAM, Philadelphia, 1994. doi: 10.1137/1.9781611971262.
[6]	G. Chen and X. Chen, A new splitting for singular linear system and Drazin inverse, J. East China Norm. Univ. Natur. sci. Ed., 3 (1996), 12-18.
[7]	L. Collatz, Functional Analysis and Numerical Mathematics, Academic, New York, 1966.
[8]	L. Jena and D. Mishra, B_D-splittings of matrices, Linear Algebra and Applications, 437 (2012), 1162-1173. doi: 10.1016/j.laa.2012.04.009.
[9]	O. L. Mangasarian, Characterization of real matrices of monotone kind, SIAM Review, 10 (1968), 439-441.
[10]	D. Mishra and K. C. Sivakumar, A dominance notion of singular matrices with applications to nonnegative generalized inverses, Linear and Multilinear Algebra, 60 (2012), 911-920. doi: 10.1080/03081087.2011.632378.
[11]	W. C. Pye, Nonnegative Drazin inverses, Linear Algebra Appl., 30 (1980), 149-153. doi: 10.1016/0024-3795(80)90190-1.
[12]	F. Szidarovszky and K. Okuguchi, A general scheme for matrices with nonnegative inverse, PU.M.A. Ser. B, 1 (1990), 109-114.
[13]	R. S. Varga, Matrix Iterative Analysis, Springer-Verlag, Berlin, 2000. doi: 10.1007/978-3-642-05156-2.
[14]	Y. Wei, Index splitting for the Drazin inverse and the singular linear system, Appl. Math. Comput., 95 (1998), 115-124. doi: 10.1016/S0096-3003(97)10098-4.
[15]	Y. Wei and H. Wu, Additional results on index splittings for Drazin inverse solutions of singular linear systems, Electron. J. Linear Algebra, 85 (2001), 83-93.
[16]	Y. Wei and H. Wu, Convergence properties of Krylov subspace methods for singular linear systems with arbitrary index, Journal of Computational and Applied Mathematics, 114 (2000), 305-318. doi: 10.1016/S0377-0427(99)90237-6.

show all references

References:

[1]	A. Ben-Israel and T. N. E. Greville, Generalized Inverses. Theory and Applications, Springer-Verlag, New York, 2003.
[2]	A. Berman and R. J. Plemmons, Cones and iterative methods for best square least squares solutions of linear systems, SIAM J. Numer. Anal., 11 (1974), 145-154.
[3]	A. Berman and R. J. Plemmons, Monotonicity and the generalized inverse, SIAM J. Appl. Math., 22 (1972), 155-161.
[4]	A. Berman and R. J. Plemmons, Matrix group monotonicity, Proceedings of the American Mathematical Society, 46 (1974), 355-359.
[5]	A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, SIAM, Philadelphia, 1994. doi: 10.1137/1.9781611971262.
[6]	G. Chen and X. Chen, A new splitting for singular linear system and Drazin inverse, J. East China Norm. Univ. Natur. sci. Ed., 3 (1996), 12-18.
[7]	L. Collatz, Functional Analysis and Numerical Mathematics, Academic, New York, 1966.
[8]	L. Jena and D. Mishra, B_D-splittings of matrices, Linear Algebra and Applications, 437 (2012), 1162-1173. doi: 10.1016/j.laa.2012.04.009.
[9]	O. L. Mangasarian, Characterization of real matrices of monotone kind, SIAM Review, 10 (1968), 439-441.
[10]	D. Mishra and K. C. Sivakumar, A dominance notion of singular matrices with applications to nonnegative generalized inverses, Linear and Multilinear Algebra, 60 (2012), 911-920. doi: 10.1080/03081087.2011.632378.
[11]	W. C. Pye, Nonnegative Drazin inverses, Linear Algebra Appl., 30 (1980), 149-153. doi: 10.1016/0024-3795(80)90190-1.
[12]	F. Szidarovszky and K. Okuguchi, A general scheme for matrices with nonnegative inverse, PU.M.A. Ser. B, 1 (1990), 109-114.
[13]	R. S. Varga, Matrix Iterative Analysis, Springer-Verlag, Berlin, 2000. doi: 10.1007/978-3-642-05156-2.
[14]	Y. Wei, Index splitting for the Drazin inverse and the singular linear system, Appl. Math. Comput., 95 (1998), 115-124. doi: 10.1016/S0096-3003(97)10098-4.
[15]	Y. Wei and H. Wu, Additional results on index splittings for Drazin inverse solutions of singular linear systems, Electron. J. Linear Algebra, 85 (2001), 83-93.
[16]	Y. Wei and H. Wu, Convergence properties of Krylov subspace methods for singular linear systems with arbitrary index, Journal of Computational and Applied Mathematics, 114 (2000), 305-318. doi: 10.1016/S0377-0427(99)90237-6.

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