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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

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 & 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.  Google Scholar

[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.  Google Scholar

[3]

A. Berman and R. J. Plemmons, Monotonicity and the generalized inverse, SIAM J. Appl. Math., 22 (1972), 155-161.  Google Scholar

[4]

A. Berman and R. J. Plemmons, Matrix group monotonicity, Proceedings of the American Mathematical Society, 46 (1974), 355-359.  Google Scholar

[5]

A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, SIAM, Philadelphia, 1994. doi: 10.1137/1.9781611971262.  Google Scholar

[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.  Google Scholar

[7]

L. Collatz, Functional Analysis and Numerical Mathematics, Academic, New York, 1966.  Google Scholar

[8]

L. Jena and D. Mishra, BD-splittings of matrices, Linear Algebra and Applications, 437 (2012), 1162-1173. doi: 10.1016/j.laa.2012.04.009.  Google Scholar

[9]

O. L. Mangasarian, Characterization of real matrices of monotone kind, SIAM Review, 10 (1968), 439-441.  Google Scholar

[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.  Google Scholar

[11]

W. C. Pye, Nonnegative Drazin inverses, Linear Algebra Appl., 30 (1980), 149-153. doi: 10.1016/0024-3795(80)90190-1.  Google Scholar

[12]

F. Szidarovszky and K. Okuguchi, A general scheme for matrices with nonnegative inverse, PU.M.A. Ser. B, 1 (1990), 109-114.  Google Scholar

[13]

R. S. Varga, Matrix Iterative Analysis, Springer-Verlag, Berlin, 2000. doi: 10.1007/978-3-642-05156-2.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

A. Ben-Israel and T. N. E. Greville, Generalized Inverses. Theory and Applications, Springer-Verlag, New York, 2003.  Google Scholar

[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.  Google Scholar

[3]

A. Berman and R. J. Plemmons, Monotonicity and the generalized inverse, SIAM J. Appl. Math., 22 (1972), 155-161.  Google Scholar

[4]

A. Berman and R. J. Plemmons, Matrix group monotonicity, Proceedings of the American Mathematical Society, 46 (1974), 355-359.  Google Scholar

[5]

A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, SIAM, Philadelphia, 1994. doi: 10.1137/1.9781611971262.  Google Scholar

[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.  Google Scholar

[7]

L. Collatz, Functional Analysis and Numerical Mathematics, Academic, New York, 1966.  Google Scholar

[8]

L. Jena and D. Mishra, BD-splittings of matrices, Linear Algebra and Applications, 437 (2012), 1162-1173. doi: 10.1016/j.laa.2012.04.009.  Google Scholar

[9]

O. L. Mangasarian, Characterization of real matrices of monotone kind, SIAM Review, 10 (1968), 439-441.  Google Scholar

[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.  Google Scholar

[11]

W. C. Pye, Nonnegative Drazin inverses, Linear Algebra Appl., 30 (1980), 149-153. doi: 10.1016/0024-3795(80)90190-1.  Google Scholar

[12]

F. Szidarovszky and K. Okuguchi, A general scheme for matrices with nonnegative inverse, PU.M.A. Ser. B, 1 (1990), 109-114.  Google Scholar

[13]

R. S. Varga, Matrix Iterative Analysis, Springer-Verlag, Berlin, 2000. doi: 10.1007/978-3-642-05156-2.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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