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Mathematical properties of the regular *-representation of matrix $*$-algebras with applications to semidefinite programming
Index-range monotonicity and index-proper splittings of matrices
1. | School of Basic Sciences, Indian Institute of Technology Bhubaneswar, Bhubaneswar - 751 013, Odisha, India, India |
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 Brow and Bran-splittings of matrices, Linear and Multilinear Algebra, 61 (2013), 35-48.
doi: 10.1080/03081087.2012.661426. |
[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. |
[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 Brow and Bran-splittings of matrices, Linear and Multilinear Algebra, 61 (2013), 35-48.
doi: 10.1080/03081087.2012.661426. |
[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. |
[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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