Index-range monotonicity and index-proper splittings of matrices

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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
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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 and 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, 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. doi: 10.1137/0711015.
[3]	A. Berman and R. J. Plemmons, Monotonicity and the generalized inverse, SIAM J. Appl. Math., 22 (1972), 155-161. doi: 10.1137/0122018.
[4]	A. Berman and R. J. Plemmons, Matrix group monotonicity, Proceedings of the American Mathematical Society, 46 (1974), 355-359. 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), 115-123. 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), 12-18.
[7]	L. Collatz, Aufgaben monotoner Art, Arch. Math., 3 (1952), 366-376. 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), 35-48. doi: 10.1080/03081087.2012.661426.
[9]	L. Jena and D. Mishra, B_D-splittings of matrices, Linear Algebra and Appl., 437 (2012), 1162-1173. 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, Berlin, 1989.
[12]	D. Mishra, "Least Elements, Matrix Splittings and Nonnegative Generalized Inverses," PhD Thesis, IIT Madras, 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), 2604-2614. 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), 45-58. 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), 167-184. doi: 10.1016/0024-3795(92)90275-F.
[16]	W. C. Pye, Nonnegative Drazin inverses, Linear Algebra Appl., 30 (1980), 149-153. doi: 10.1016/0024-3795(80)90190-1.
[17]	A. Schrijver, "Theory of Linear and Integer Programming," John Wiley & Sons Ltd., Chichester, 1986.
[18]	Y. Song, Comparisons of nonnegative splittings of matrices, Linear Algebra Appl., 154-156 (1991), 453-455. doi: 10.1016/0024-3795(91)90388-D.
[19]	R. S. Varga, "Matrix Iterative Analysis," Springer-Verlag, Berlin, 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), 115-124. doi: 10.1016/S0096-3003(97)10098-4.
[21]	Z. I. Woźnicki, Matrix splitting principles, Novi Sad J. Math., 28 (1998), 197-209.
[22]	Z. I. Woźnicki, Nonnegative splitting theory, Japan J. Industr. Appl. Math., 11 (1994), 289-342. doi: 10.1007/BF03167226.

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. doi: 10.1137/0711015.
[3]	A. Berman and R. J. Plemmons, Monotonicity and the generalized inverse, SIAM J. Appl. Math., 22 (1972), 155-161. doi: 10.1137/0122018.
[4]	A. Berman and R. J. Plemmons, Matrix group monotonicity, Proceedings of the American Mathematical Society, 46 (1974), 355-359. 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), 115-123. 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), 12-18.
[7]	L. Collatz, Aufgaben monotoner Art, Arch. Math., 3 (1952), 366-376. 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), 35-48. doi: 10.1080/03081087.2012.661426.
[9]	L. Jena and D. Mishra, B_D-splittings of matrices, Linear Algebra and Appl., 437 (2012), 1162-1173. 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, Berlin, 1989.
[12]	D. Mishra, "Least Elements, Matrix Splittings and Nonnegative Generalized Inverses," PhD Thesis, IIT Madras, 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), 2604-2614. 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), 45-58. 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), 167-184. doi: 10.1016/0024-3795(92)90275-F.
[16]	W. C. Pye, Nonnegative Drazin inverses, Linear Algebra Appl., 30 (1980), 149-153. doi: 10.1016/0024-3795(80)90190-1.
[17]	A. Schrijver, "Theory of Linear and Integer Programming," John Wiley & Sons Ltd., Chichester, 1986.
[18]	Y. Song, Comparisons of nonnegative splittings of matrices, Linear Algebra Appl., 154-156 (1991), 453-455. doi: 10.1016/0024-3795(91)90388-D.
[19]	R. S. Varga, "Matrix Iterative Analysis," Springer-Verlag, Berlin, 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), 115-124. doi: 10.1016/S0096-3003(97)10098-4.
[21]	Z. I. Woźnicki, Matrix splitting principles, Novi Sad J. Math., 28 (1998), 197-209.
[22]	Z. I. Woźnicki, Nonnegative splitting theory, Japan J. Industr. Appl. Math., 11 (1994), 289-342. doi: 10.1007/BF03167226.

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