(R, S)-(Skew) Symmetric Solutions to Matrix Equation AXB = C over Quaternions

(R, S)-(Skew) Symmetric Solutions to Matrix Equation AXB = C over Quaternions Ruopeng Liao School of Computer Science and Engineering, Faculty of Innovation Engineering, Macau University of Science and Technology, Avenida Wai Long, TaiPa, Macau 999078, China http://orcid.org/0000-0002-8303-1966 Xin Liu Macau Institute of Systems Engineering, Faculty of Innovation Engineering, Macau University of Science and Technology, Avenida Wai Long, TaiPa, Macau 999078, China http://orcid.org/0000-0001-9646-4448 Sujuan Long School of Mathematics and Data Science, Minjiang University, Fujian 350108, China Yang Zhang Department of Mathematics, University of Manitoba, Winnipeg, MB R3T 2N2, Canada

(R,S)-(skew) symmetric matrices have numerous applications in civil engineering, information theory, numerical analysis, etc. In this paper, we deal with the (R,S)-(skew) symmetric solutions to the quaternion matrix equation AXB=C. We use a real representation Aτ to obtain the necessary and sufficient conditions for AXB=C to have (R,S)-(skew) symmetric solutions and derive the solutions when it is consistent. We also derive the least-squares (R,S)-(skew) symmetric solution to the above matrix equation.
01 18 2024 323 math12020323 Macao Science and Technology Development Fund http://dx.doi.org/10.13039/ 0013/2021/ITP National Natural Science Foundation of China http://dx.doi.org/10.13039/ 12371023 12271338 12001259 Natural Sciences and Engineering Research Council of Canada http://dx.doi.org/10.13039/ RGPIN 2020-06746 Joint Research and Development Fund ofWuyi University, Hong Kong and Macao http://dx.doi.org/10.13039/ 2019WGALH20 MUST Faculty Research http://dx.doi.org/10.13039/ FRG-22-073-FIE Science Foundation of Fujian Province http://dx.doi.org/10.13039/ 2020J01846 Research Foundation of Minjiang University for the Introduction of Talents http://dx.doi.org/10.13039/ MJY17006 https://creativecommons.org/licenses/by/4.0/ 10.3390/math12020323 https://www.mdpi.com/2227-7390/12/2/323 https://www.mdpi.com/2227-7390/12/2/323/pdf Chen New structure-preserving quaternion QR decomposition method for color image blind watermarking Signal Process. 2021 10.1016/j.sigpro.2021.108088 185 108088 10.1109/IJCNN.2018.8489651 Gaudet, C.J., and Maida, A.S. (2018, January 8–13). Deep quaternion networks. Proceedings of the 2018 International Joint Conference on Neural Networks, Rio de Janeiro, Brazil. He A real quaternion matrix equation with applications Linear Multilinear Algebra 2013 10.1080/03081087.2012.703192 61 725 Mandic A quaternion gradient operator and its applications IEEE Signal Process. Lett. 2011 10.1109/LSP.2010.2091126 18 47 Yang Spacecraft attitude determination and control: Quaternion based method Annu. Rev. Control 2012 10.1016/j.arcontrol.2012.09.003 36 198 Zhu Quaternion convolutional neural networks Proc. Eur. Conf. Comput. Vis. 2018 7 631 Dehghan Matrix equations over (R, S)-symmetric and (R, S)-skew symmetric matrices Comput. Math. Appl. 2010 10.1016/j.camwa.2010.03.052 59 3583 Trench Minimization problems for (R, S)-symmetric and (R, S)-skew symmetric matrices Linear Algebra Appl. 2014 10.1016/j.laa.2004.03.035 389 23 Chen Generalized Reflexive Matrices: Special Properties and Applications SIAM J. Matrix Anal. Appl. 1998 10.1137/S0895479895288759 19 140 Lv The iterative algorithm for solving a class of generalized coupled Sylvester-transpose equations over centrosymmetric or anti-centrosymmetric matrix Int. J. Comput. Math. 2019 10.1080/00207160.2018.1449946 96 1576 Wang A system of real quaternion matrix equations with applications Linear Algebra Appl. 2009 10.1016/j.laa.2009.02.010 431 2291 Xu Inverse problems for (R, S)-symmetric matrices in structural dynamic model updating Comput. Math. Appl. 2016 10.1016/j.camwa.2016.01.026 71 1074 Xie Iterative method to solve the generalized coupled Sylvester-transpose linear matrix equations over reflexive or anti-reflexive matrix Comput. Math. Appl. 2014 10.1016/j.camwa.2014.04.012 67 2071 Yuan Generalized reflexive solutions of the matrix equation AXB = D and an associated optimal approximation problem Comput. Math. Appl. 2008 10.1016/j.camwa.2008.03.015 56 1643 Zhang The (P, Q)-(skew) symmetric extremal rank solutions to a system of quaternion matrix equations Appl. Math. Comput. 2011 217 9286 Peng The reflexive and anti-reflexive solutions of the matrix equation AX = B Linear Algebra Appl. 2003 10.1016/S0024-3795(03)00607-4 375 147 Wang P-(skew) symmetric common solutions to a pair of quaternion matrix equations Appl. Math. Comput. 2008 195 721 Zhou Least-square solutions for inverse problems of centrosymmetric matrices Comput. Math. Appl. 2003 10.1016/S0898-1221(03)00137-8 45 1581 Jiang Algebraic methods for diagonalization of a quaternion matrix in quaternionic quantum theory J. Math. Phys. 2005 10.1063/1.1896386 46 783 Liu Consistency of quaternion matrix equations AX⋆−XB = C,X−AX⋆B=C Electron. J. Linear Algebra 2019 10.13001/1081-3810.3950 35 394 Rodman, L. (2014). Princeton Series in Applied Mathematics, Princeton University Press. Ben-Israel, A., and Greville, T.N.E. (1974). Generalized Inverses: Theory and Applications, Wiley.