×

Fuzzy symmetric solutions of fuzzy matrix equations. (English) Zbl 1250.15023

Summary: The fuzzy symmetric solution of fuzzy matrix equation \(A\tilde{X} = \tilde{B}\), in which \(A\) is a crisp \(m \times m\) nonsingular matrix and \(\tilde{B}\) is an \(m \times n\) fuzzy numbers matrix with nonzero spreads, is investigated. The fuzzy matrix equation is converted to a fuzzy system of linear equations according to the Kronecker product of matrices. From solving the fuzzy linear system, three types of fuzzy symmetric solutions of the fuzzy matrix equation are derived. Finally, two examples are given to illustrate the proposed method.

MSC:

15A24 Matrix equations and identities
15B15 Fuzzy matrices

Software:

INTOPT_90
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] L. A. Zadeh, “The concept of a linguistic variable and its application to approximate reasoning-I,” Information Sciences, vol. 8, no. 3, pp. 199-249, 1975. · Zbl 0397.68071 · doi:10.1016/0020-0255(75)90036-5
[2] D. Dubois and H. Prade, “Operations on fuzzy numbers,” Journal of Systems Science, vol. 9, no. 6, pp. 613-626, 1978. · Zbl 0383.94045 · doi:10.1080/00207727808941724
[3] S. Nahmias, “Fuzzy variables,” Fuzzy Sets and Systems, vol. 1, no. 2, pp. 97-110, 1978. · Zbl 0383.03038 · doi:10.1016/0165-0114(78)90011-8
[4] M. L. Puri and D. A. Ralescu, “Differentials of fuzzy functions,” Journal of Mathematical Analysis and Applications, vol. 91, no. 2, pp. 552-558, 1983. · Zbl 0528.54009 · doi:10.1016/0022-247X(83)90169-5
[5] R. Goetschel and W. Voxman, “Elementary fuzzy calculus,” Fuzzy Sets and Systems, vol. 18, no. 1, pp. 31-43, 1986. · Zbl 0626.26014 · doi:10.1016/0165-0114(86)90026-6
[6] C.-X. Wu and M. Ming, “Embedding problem of fuzzy number space: part I,” Fuzzy Sets and Systems, vol. 44, no. 1, pp. 33-38, 1991. · Zbl 0757.46066 · doi:10.1016/0165-0114(91)90030-T
[7] C. X. Wu and M. Ming, “Embedding problem of fuzzy number space: part III,” Fuzzy Sets and Systems, vol. 46, no. 2, pp. 281-286, 1992. · Zbl 0774.54003 · doi:10.1016/0165-0114(92)90142-Q
[8] M. Friedman, M. Ming, and A. Kandel, “Fuzzy linear systems,” Fuzzy Sets and Systems, vol. 96, no. 2, pp. 201-209, 1998. · Zbl 0929.15004 · doi:10.1016/S0165-0114(96)00270-9
[9] M. Ma, M. Friedman, and A. Kandel, “Duality in fuzzy linear systems,” Fuzzy Sets and Systems, vol. 109, no. 1, pp. 55-58, 2000. · Zbl 0945.15002 · doi:10.1016/S0165-0114(98)00102-X
[10] T. Allahviranloo, M. Friedman, M. Ma, and A. Kandel, “A comment on fuzzy linear systems (multiple letters),” Fuzzy Sets and Systems, vol. 140, no. 3, pp. 559-561, 2003. · Zbl 1050.15003 · doi:10.1016/S0165-0114(03)00139-8
[11] T. Allahviranloo, “Numerical methods for fuzzy system of linear equations,” Applied Mathematics and Computation, vol. 155, no. 2, pp. 493-502, 2004. · Zbl 1067.65040 · doi:10.1016/S0096-3003(03)00793-8
[12] T. Allahviranloo, “Successive over relaxation iterative method for fuzzy system of linear equations,” Applied Mathematics and Computation, vol. 162, no. 1, pp. 189-196, 2005. · Zbl 1062.65037 · doi:10.1016/j.amc.2003.12.085
[13] T. Allahviranloo, “The Adomian decomposition method for fuzzy system of linear equations,” Applied Mathematics and Computation, vol. 163, no. 2, pp. 553-563, 2005. · Zbl 1069.65025 · doi:10.1016/j.amc.2004.02.020
[14] S. Abbasbandy, R. Ezzati, and A. Jafarian, “LU decomposition method for solving fuzzy system of linear equations,” Applied Mathematics and Computation, vol. 172, no. 1, pp. 633-643, 2006. · Zbl 1088.65023 · doi:10.1016/j.amc.2005.02.018
[15] S. Abbasbandy, A. Jafarian, and R. Ezzati, “Conjugate gradient method for fuzzy symmetric positive definite system of linear equations,” Applied Mathematics and Computation, vol. 171, no. 2, pp. 1184-1191, 2005. · Zbl 1121.65311 · doi:10.1016/j.amc.2005.01.110
[16] S. Abbasbandy, M. Otadi, and M. Mosleh, “Minimal solution of general dual fuzzy linear systems,” Chaos, Solitons and Fractals, vol. 37, no. 4, pp. 1113-1124, 2008. · Zbl 1146.15002 · doi:10.1016/j.chaos.2006.10.045
[17] B. Asady, S. Abbasbandy, and M. Alavi, “Fuzzy general linear systems,” Applied Mathematics and Computation, vol. 169, no. 1, pp. 34-40, 2005. · Zbl 1119.65325 · doi:10.1016/j.amc.2004.10.042
[18] K. Wang and B. Zheng, “Inconsistent fuzzy linear systems,” Applied Mathematics and Computation, vol. 181, no. 2, pp. 973-981, 2006. · Zbl 1122.15004 · doi:10.1016/j.amc.2006.02.019
[19] B. Zheng and K. Wang, “General fuzzy linear systems,” Applied Mathematics and Computation, vol. 181, no. 2, pp. 1276-1286, 2006. · Zbl 1122.15005 · doi:10.1016/j.amc.2006.02.027
[20] M. Dehghan and B. Hashemi, “Iterative solution of fuzzy linear systems,” Applied Mathematics and Computation, vol. 175, no. 1, pp. 645-674, 2006. · Zbl 1137.65336 · doi:10.1016/j.amc.2005.07.033
[21] M. Dehghan, B. Hashemi, and M. Ghatee, “Solution of the fully fuzzy linear systems using iterative techniques,” Chaos, Solitons and Fractals, vol. 34, no. 2, pp. 316-336, 2007. · Zbl 1144.65021 · doi:10.1016/j.chaos.2006.03.085
[22] X. B. Guo and Z. T. Gong, “Block gaussian elimination methods for fuzzy matrix equations,” International Journal of Pure and Applied Mathematics, vol. 58, no. 2, pp. 157-168, 2010. · Zbl 1192.65034
[23] X. B. Guo and Z. T. Gong, “Undetermined coefficients method for solving semi-fuzzy matrix equations,” in Proceedings of 8th InternationalConference on Machine Learning and Cybernetics (ICMLC ’10), pp. 596-600, Qingdao, China, July 2010. · doi:10.1109/ICMLC.2010.5580544
[24] Z. T. Gong and X. B. Guo, “Inconsistent fuzzy matrix equations and its fuzzy least squares solutions,” Applied Mathematical Modelling, vol. 35, no. 3, pp. 1456-1469, 2011. · Zbl 1211.15038 · doi:10.1016/j.apm.2010.09.022
[25] T. Allahviranloo and S. Salahshour, “Fuzzy symmetric solutions of fuzzy linear systems,” Journal of Computational and Applied Mathematics, vol. 235, no. 16, pp. 4545-4553, 2011. · Zbl 1220.65032 · doi:10.1016/j.cam.2010.02.042
[26] T. Allahviranloo, S. Salahshour, and M. Khezerloo, “Maximal- and minimal symmetric solutions of fully fuzzy linear systems,” Journal of Computational and Applied Mathematics, vol. 235, no. 16, pp. 4652-4662, 2011. · Zbl 1220.65033 · doi:10.1016/j.cam.2010.05.009
[27] X. D. Zhang, Matrix Analysis and Applications, Tsinghua and Springer Press, Beijing, China, 2004.
[28] R. B. Kearfott, Regorous Global Search: Continuous Problems, Kluwer Academic Publishers, Amsterdam, The Netherlands, 1996. · Zbl 0876.90082
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.