Index-range monotonicity and index-proper splittings of matrices

Previous Article
Mathematical properties of the regular *-representation of matrix $*$-algebras with applications to semidefinite programming
NACO Home
This Issue
Next Article

2013, 3(2): 379-388. doi: 10.3934/naco.2013.3.379

Index-range monotonicity and index-proper splittings of matrices

Litismita Jena ^1, and Sabyasachi Pani ^1,

1.	School of Basic Sciences, Indian Institute of Technology Bhubaneswar, Bhubaneswar - 751 013, Odisha, India, India

Received April 2012 Revised January 2013 Published April 2013

Abstract
Related Papers

Index-range monotonicity is proposed, and some characterizations of this notion are obtained. Then different convergence and comparison theorems are presented using several new subclasses of index-proper splittings.

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

Mathematics Subject Classification: Primary: 15A09, 65F15, 65F2.

Citation: Litismita Jena, Sabyasachi Pani. Index-range monotonicity and index-proper splittings of matrices. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 379-388. doi: 10.3934/naco.2013.3.379

References:

[1]	A. Ben-Israel and T. N. E. Greville, "Generalized Inverses, Theory and Applications,", Springer-Verlag, (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. doi: 10.1137/0711015.
[3]	A. Berman and R. J. Plemmons, Monotonicity and the generalized inverse,, SIAM J. Appl. Math., 22 (1972), 155. doi: 10.1137/0122018.
[4]	A. Berman and R. J. Plemmons, Matrix group monotonicity,, Proceedings of the American Mathematical Society, 46 (1974), 355. doi: 10.1090/S0002-9939-1974-0352116-0.
[5]	A. Berman and R. J. Plemmons, Eight types of matrix monotonicity,, Linear Algebra and Appl., 13 (1976), 115. doi: 10.1016/0024-3795(76)90049-5.
[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.
[7]	L. Collatz, Aufgaben monotoner Art,, Arch. Math., 3 (1952), 366. doi: 10.1007/BF01899376.
[8]	L. Jena and D. Mishra, Comparison theorems for B_row and B_ran-splittings of matrices,, Linear and Multilinear Algebra, 61 (2013), 35. doi: 10.1080/03081087.2012.661426.
[9]	L. Jena and D. Mishra, B_D-splittings of matrices,, Linear Algebra and Appl., 437 (2012), 1162. doi: 10.1016/j.laa.2012.04.009.
[10]	L. Jena and S. Pani, Interval Drazin monotonicity of matrices,, Revised version submitted to Vietnam Journal of Mathematics., ().
[11]	M. A. Krasnosel'skij, A. Je. Lifshits and A. V. Sobolev, "Positive Linear Systems,", Heldermann Verlag, (1989).
[12]	D. Mishra, "Least Elements, Matrix Splittings and Nonnegative Generalized Inverses,", PhD Thesis, (2012).
[13]	D. Mishra and K. C. Sivakumar, Generalizations of matrix monotonicity and their relationships with certain subclasses of proper splittings,, Linear Algebra Appl., 436 (2012), 2604. doi: 10.1016/j.laa.2011.11.016.
[14]	J. E. Peris, A new characterization of inverse-positive matrices,, Linear Algebra Appl., 154/156 (1991), 45. doi: 10.1016/0024-3795(91)90372-4.
[15]	J. E. Peris and B. Subizas, A characterization of weak-monotone matrices,, Linear Algebra Appl., 166 (1992), 167. doi: 10.1016/0024-3795(92)90275-F.
[16]	W. C. Pye, Nonnegative Drazin inverses,, Linear Algebra Appl., 30 (1980), 149. doi: 10.1016/0024-3795(80)90190-1.
[17]	A. Schrijver, "Theory of Linear and Integer Programming,", John Wiley & Sons Ltd., (1986).
[18]	Y. Song, Comparisons of nonnegative splittings of matrices,, Linear Algebra Appl., 154-156 (1991), 154. doi: 10.1016/0024-3795(91)90388-D.
[19]	R. S. Varga, "Matrix Iterative Analysis,", Springer-Verlag, (2000). doi: 10.1007/978-3-642-05156-2.
[20]	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.
[21]	Z. I. Woźnicki, Matrix splitting principles,, Novi Sad J. Math., 28 (1998), 197.
[22]	Z. I. Woźnicki, Nonnegative splitting theory,, Japan J. Industr. Appl. Math., 11 (1994), 289. doi: 10.1007/BF03167226.

show all references

References:

[1]	A. Ben-Israel and T. N. E. Greville, "Generalized Inverses, Theory and Applications,", Springer-Verlag, (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. doi: 10.1137/0711015.
[3]	A. Berman and R. J. Plemmons, Monotonicity and the generalized inverse,, SIAM J. Appl. Math., 22 (1972), 155. doi: 10.1137/0122018.
[4]	A. Berman and R. J. Plemmons, Matrix group monotonicity,, Proceedings of the American Mathematical Society, 46 (1974), 355. doi: 10.1090/S0002-9939-1974-0352116-0.
[5]	A. Berman and R. J. Plemmons, Eight types of matrix monotonicity,, Linear Algebra and Appl., 13 (1976), 115. doi: 10.1016/0024-3795(76)90049-5.
[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.
[7]	L. Collatz, Aufgaben monotoner Art,, Arch. Math., 3 (1952), 366. doi: 10.1007/BF01899376.
[8]	L. Jena and D. Mishra, Comparison theorems for B_row and B_ran-splittings of matrices,, Linear and Multilinear Algebra, 61 (2013), 35. doi: 10.1080/03081087.2012.661426.
[9]	L. Jena and D. Mishra, B_D-splittings of matrices,, Linear Algebra and Appl., 437 (2012), 1162. doi: 10.1016/j.laa.2012.04.009.
[10]	L. Jena and S. Pani, Interval Drazin monotonicity of matrices,, Revised version submitted to Vietnam Journal of Mathematics., ().
[11]	M. A. Krasnosel'skij, A. Je. Lifshits and A. V. Sobolev, "Positive Linear Systems,", Heldermann Verlag, (1989).
[12]	D. Mishra, "Least Elements, Matrix Splittings and Nonnegative Generalized Inverses,", PhD Thesis, (2012).
[13]	D. Mishra and K. C. Sivakumar, Generalizations of matrix monotonicity and their relationships with certain subclasses of proper splittings,, Linear Algebra Appl., 436 (2012), 2604. doi: 10.1016/j.laa.2011.11.016.
[14]	J. E. Peris, A new characterization of inverse-positive matrices,, Linear Algebra Appl., 154/156 (1991), 45. doi: 10.1016/0024-3795(91)90372-4.
[15]	J. E. Peris and B. Subizas, A characterization of weak-monotone matrices,, Linear Algebra Appl., 166 (1992), 167. doi: 10.1016/0024-3795(92)90275-F.
[16]	W. C. Pye, Nonnegative Drazin inverses,, Linear Algebra Appl., 30 (1980), 149. doi: 10.1016/0024-3795(80)90190-1.
[17]	A. Schrijver, "Theory of Linear and Integer Programming,", John Wiley & Sons Ltd., (1986).
[18]	Y. Song, Comparisons of nonnegative splittings of matrices,, Linear Algebra Appl., 154-156 (1991), 154. doi: 10.1016/0024-3795(91)90388-D.
[19]	R. S. Varga, "Matrix Iterative Analysis,", Springer-Verlag, (2000). doi: 10.1007/978-3-642-05156-2.
[20]	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.
[21]	Z. I. Woźnicki, Matrix splitting principles,, Novi Sad J. Math., 28 (1998), 197.
[22]	Z. I. Woźnicki, Nonnegative splitting theory,, Japan J. Industr. Appl. Math., 11 (1994), 289. doi: 10.1007/BF03167226.

[1]	Chinmay Kumar Giri. Index-proper nonnegative splittings of matrices. Numerical Algebra, Control & Optimization, 2016, 6 (2) : 103-113. doi: 10.3934/naco.2016002
[2]	Haifeng Ma, Xiaoshuang Gao. Further results on the perturbation estimations for the Drazin inverse. Numerical Algebra, Control & Optimization, 2018, 8 (4) : 493-503. doi: 10.3934/naco.2018031
[3]	Ricardo Weder, Dimitri Yafaev. Inverse scattering at a fixed energy for long-range potentials. Inverse Problems & Imaging, 2007, 1 (1) : 217-224. doi: 10.3934/ipi.2007.1.217
[4]	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
[5]	S. Yu. Pilyugin. Inverse shadowing by continuous methods. Discrete & Continuous Dynamical Systems - A, 2002, 8 (1) : 29-38. doi: 10.3934/dcds.2002.8.29
[6]	Colin Guillarmou, Antônio Sá Barreto. Inverse problems for Einstein manifolds. Inverse Problems & Imaging, 2009, 3 (1) : 1-15. doi: 10.3934/ipi.2009.3.1
[7]	Sergei Avdonin, Pavel Kurasov. Inverse problems for quantum trees. Inverse Problems & Imaging, 2008, 2 (1) : 1-21. doi: 10.3934/ipi.2008.2.1
[8]	Maciej Zworski. A remark on inverse problems for resonances. Inverse Problems & Imaging, 2007, 1 (1) : 225-227. doi: 10.3934/ipi.2007.1.225
[9]	Victor Isakov, Joseph Myers. On the inverse doping profile problem. Inverse Problems & Imaging, 2012, 6 (3) : 465-486. doi: 10.3934/ipi.2012.6.465
[10]	Fabrizio Colombo, Irene Sabadini, Frank Sommen. The inverse Fueter mapping theorem. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1165-1181. doi: 10.3934/cpaa.2011.10.1165
[11]	Guanghui Hu, Peijun Li, Xiaodong Liu, Yue Zhao. Inverse source problems in electrodynamics. Inverse Problems & Imaging, 2018, 12 (6) : 1411-1428. doi: 10.3934/ipi.2018059
[12]	Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems & Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271
[13]	Leonardo Marazzi. Inverse scattering on conformally compact manifolds. Inverse Problems & Imaging, 2009, 3 (3) : 537-550. doi: 10.3934/ipi.2009.3.537
[14]	Guillaume Bal, Alexandre Jollivet. Stability estimates in stationary inverse transport. Inverse Problems & Imaging, 2008, 2 (4) : 427-454. doi: 10.3934/ipi.2008.2.427
[15]	Janne M.J. Huttunen, J. P. Kaipio. Approximation errors in nonstationary inverse problems. Inverse Problems & Imaging, 2007, 1 (1) : 77-93. doi: 10.3934/ipi.2007.1.77
[16]	Henk Bruin, Sonja Štimac. On isotopy and unimodal inverse limit spaces. Discrete & Continuous Dynamical Systems - A, 2012, 32 (4) : 1245-1253. doi: 10.3934/dcds.2012.32.1245
[17]	Nguyen Huy Tuan, Mokhtar Kirane, Long Dinh Le, Van Thinh Nguyen. On an inverse problem for fractional evolution equation. Evolution Equations & Control Theory, 2017, 6 (1) : 111-134. doi: 10.3934/eect.2017007
[18]	Masoumeh Dashti, Stephen Harris, Andrew Stuart. Besov priors for Bayesian inverse problems. Inverse Problems & Imaging, 2012, 6 (2) : 183-200. doi: 10.3934/ipi.2012.6.183
[19]	Russell Johnson, Luca Zampogni. On the inverse Sturm-Liouville problem. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 405-428. doi: 10.3934/dcds.2007.18.405
[20]	Mikko Orispää, Markku Lehtinen. Fortran linear inverse problem solver. Inverse Problems & Imaging, 2010, 4 (3) : 485-503. doi: 10.3934/ipi.2010.4.485

Impact Factor:

American Institute of Mathematical Sciences