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

Matrix group monotonicity using a dominance notion

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.
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, (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. Google Scholar

[3]

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

[4]

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

[5]

A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences,, SIAM, (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. Google Scholar

[7]

L. Collatz, Functional Analysis and Numerical Mathematics,, Academic, (1966). Google Scholar

[8]

L. Jena and D. Mishra, BD-splittings of matrices,, Linear Algebra and Applications, 437 (2012), 1162. 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. 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. doi: 10.1080/03081087.2011.632378. Google Scholar

[11]

W. C. Pye, Nonnegative Drazin inverses,, Linear Algebra Appl., 30 (1980), 149. 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. Google Scholar

[13]

R. S. Varga, Matrix Iterative Analysis,, Springer-Verlag, (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. 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. 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. 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, (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. Google Scholar

[3]

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

[4]

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

[5]

A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences,, SIAM, (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. Google Scholar

[7]

L. Collatz, Functional Analysis and Numerical Mathematics,, Academic, (1966). Google Scholar

[8]

L. Jena and D. Mishra, BD-splittings of matrices,, Linear Algebra and Applications, 437 (2012), 1162. 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. 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. doi: 10.1080/03081087.2011.632378. Google Scholar

[11]

W. C. Pye, Nonnegative Drazin inverses,, Linear Algebra Appl., 30 (1980), 149. 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. Google Scholar

[13]

R. S. Varga, Matrix Iterative Analysis,, Springer-Verlag, (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. 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. 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. doi: 10.1016/S0377-0427(99)90237-6. Google Scholar

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