American Institute of Mathematical Sciences

Advanced
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

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, 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. doi: 10.1137/0711015.  Google Scholar

[3]

A. Berman and R. J. Plemmons, Monotonicity and the generalized inverse, SIAM J. Appl. Math., 22 (1972), 155-161. doi: 10.1137/0122018.  Google Scholar

[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.  Google Scholar

[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.  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, Aufgaben monotoner Art, Arch. Math., 3 (1952), 366-376. doi: 10.1007/BF01899376.  Google Scholar

[8]

L. Jena and D. Mishra, Comparison theorems for Brow and Bran-splittings of matrices, Linear and Multilinear Algebra, 61 (2013), 35-48. doi: 10.1080/03081087.2012.661426.  Google Scholar

[9]

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

[10]

L. Jena and S. Pani, Interval Drazin monotonicity of matrices,, Revised version submitted to Vietnam Journal of Mathematics., ().   Google Scholar

[11]

M. A. Krasnosel'skij, A. Je. Lifshits and A. V. Sobolev, "Positive Linear Systems," Heldermann Verlag, Berlin, 1989.  Google Scholar

[12]

D. Mishra, "Least Elements, Matrix Splittings and Nonnegative Generalized Inverses," PhD Thesis, IIT Madras, 2012. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[17]

A. Schrijver, "Theory of Linear and Integer Programming," John Wiley & Sons Ltd., Chichester, 1986.  Google Scholar

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[21]

Z. I. Woźnicki, Matrix splitting principles, Novi Sad J. Math., 28 (1998), 197-209.  Google Scholar

[22]

Z. I. Woźnicki, Nonnegative splitting theory, Japan J. Industr. Appl. Math., 11 (1994), 289-342. doi: 10.1007/BF03167226.  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. doi: 10.1137/0711015.  Google Scholar

[3]

A. Berman and R. J. Plemmons, Monotonicity and the generalized inverse, SIAM J. Appl. Math., 22 (1972), 155-161. doi: 10.1137/0122018.  Google Scholar

[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.  Google Scholar

[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.  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, Aufgaben monotoner Art, Arch. Math., 3 (1952), 366-376. doi: 10.1007/BF01899376.  Google Scholar

[8]

L. Jena and D. Mishra, Comparison theorems for Brow and Bran-splittings of matrices, Linear and Multilinear Algebra, 61 (2013), 35-48. doi: 10.1080/03081087.2012.661426.  Google Scholar

[9]

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

[10]

L. Jena and S. Pani, Interval Drazin monotonicity of matrices,, Revised version submitted to Vietnam Journal of Mathematics., ().   Google Scholar

[11]

M. A. Krasnosel'skij, A. Je. Lifshits and A. V. Sobolev, "Positive Linear Systems," Heldermann Verlag, Berlin, 1989.  Google Scholar

[12]

D. Mishra, "Least Elements, Matrix Splittings and Nonnegative Generalized Inverses," PhD Thesis, IIT Madras, 2012. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[17]

A. Schrijver, "Theory of Linear and Integer Programming," John Wiley & Sons Ltd., Chichester, 1986.  Google Scholar

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[21]

Z. I. Woźnicki, Matrix splitting principles, Novi Sad J. Math., 28 (1998), 197-209.  Google Scholar

[22]

Z. I. Woźnicki, Nonnegative splitting theory, Japan J. Industr. Appl. Math., 11 (1994), 289-342. doi: 10.1007/BF03167226.  Google Scholar

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