An efficient real representation method for least squares problem of the quaternion constrained matrix equation AXB + CY D = E

References

• R.K. Cavin Iii and S.P. Bhattacharyya, Robust and well-conditioned eigenstructure assignment via Sylvester equation, Optim. Control Appl. Meth. 4 (1983), pp. 205–212. doi: 10.1002/oca.4660040302
   Web of Science ®Google Scholar
• G.L. Chen, J.W. Xia, and G.M. Zhuang, Improved passivity analysis for neural networks with Markovian jumping parameters and interval time-varying delays, Neurocomputing 155 (2015), pp. 253–260. doi: 10.1016/j.neucom.2014.12.023
   Web of Science ®Google Scholar
• G.R. Duan, Eigenstructure assignment and response analysis in descriptor linear systems with state feedback control, Int. J. Control 69 (1998), pp. 663–694. doi: 10.1080/002071798222622
   Web of Science ®Google Scholar
• Z.H. He, Q.W. Wang, and Y. Zhang, Simultaneous decomposition of quaternion matrices involving η-Hermicity with applications, Appl. Math. Comput. 298 (2017), pp. 13–35. doi: 10.1016/j.amc.2016.11.002
   Web of Science ®Google Scholar
• Z.H. He, Q.W. Wang, and Y. Zhang, A simultaneous decomposition for seven matrices with application, J. Comput. Appl. Math. 349 (2019), pp. 93–113. doi: 10.1016/j.cam.2018.09.001
   Web of Science ®Google Scholar
• Z.G. Jia, M.S. Wei, and S.T. Ling, A new structure-preserving method for quaternion Hermitian eigenvalue problems, J. Comput. Appl. Math. 239 (2013), pp. 12–24. doi: 10.1016/j.cam.2012.09.018
   Web of Science ®Google Scholar
• S.T. Ling, Z.G. Jia, X. Lu, and B. Yang, Matrix LSQR algorithm for structured solutions to quaternionic least squares problem, Comput. Math. Appl. 77 (2019), pp. 830–845. doi: 10.1016/j.camwa.2018.10.023
   Web of Science ®Google Scholar
• X. Liu, The η-anti-Hermitian solution to some classic matrix equations, Appl. Math. Comput. 320 (2018), pp. 264–270. doi: 10.1016/j.amc.2017.09.033
   Web of Science ®Google Scholar
• X.R. Nie, Q.W. Wang, and Y. Zhang, A system of matrix equations over the quaternion algebra with applications, Algebra Colloq. 24 (2017), pp. 233–253. doi: 10.1142/S100538671700013X
   Web of Science ®Google Scholar
• A. Rehman, Q.W. Wang, and Z.H. He, Solution to a system of real quaternion matrix equations encompassing η-Hermicity, Appl. Math. Comput. 265 (2015), pp. 945–957. doi: 10.1016/j.amc.2015.05.104
   Web of Science ®Google Scholar
• A. Rehman, Q.W. Wang, I. Ali, M. Akram, and M.O. Ahmad, A constraint system of generalized Sylvester quaternion matrix equations, Adv. Appl. Clifford Al. 27 (2017), pp. 3183–3196. doi: 10.1007/s00006-017-0803-1
   Web of Science ®Google Scholar
• C.C. Took and D.P. Mandic, The quaternion LMS algorithm for adaptive filtering of hypercomplex real world processes, IEEE Trans. Signal Process. 57 (2009), pp. 1316–1327. doi: 10.1109/TSP.2008.2010600
   Web of Science ®Google Scholar
• C.C. Took and D.P. Mandic, Quaternion-valued stochastic gradient-based adaptive IIR filtering, IEEE Trans. Signal Process. 58 (2010), pp. 3895–3901. doi: 10.1109/TSP.2010.2047719
   Web of Science ®Google Scholar
• C.C. Took and D.P. Mandic, Augmented second-order statistics of quaternion random signals, Signal Process. 91 (2011), pp. 214–224. doi: 10.1016/j.sigpro.2010.06.024
   Web of Science ®Google Scholar
• C.C. Took, D.P. Mandic, and F.Z. Zhang, On the unitary diagonalization of a special class of quaternion matrices, Appl. Math. Lett. 24 (2011), pp. 1806–1809. doi: 10.1016/j.aml.2011.04.038
   Web of Science ®Google Scholar
• M.H. Wang, Algorithm Q-LSQR for the least squares problem in quaternionic quantum theory, Comput. Phys. Commun. 181 (2010), pp. 1047–1050. doi: 10.1016/j.cpc.2010.02.011
   Web of Science ®Google Scholar
• Q.W. Wang, X.X. Yang, and S.F. Yuan, The least square solution with the least norm to a system of quaternion matrix equations, Iran J. Sci. Technol. Trans. Sci. 42 (2018), pp. 1317–1325. doi: 10.1007/s40995-017-0472-x
   Web of Science ®Google Scholar
• Q.W. Wang, Z.H. He, and Y. Zhang, Constrained two-side dcoupled Sylvester-type quaternion matrix equations, Automatica 101 (2019), pp. 207–213. doi: 10.1016/j.automatica.2018.12.001
   Web of Science ®Google Scholar
• A.G. Wu, G.R. Duan, and B. Zhou, Solution to generalized Sylvester matrix equations, IEEE Trans. Automat. Control 53 (2008), pp. 811–815. doi: 10.1109/TAC.2008.919562
   Web of Science ®Google Scholar
• S.F. Yuan and Q.W. Wang, Two special kinds of least squares solutions for the quaternion matrix equation AXB + CXD = E, Electron. J. Linear Algebra 23 (2012), pp. 257–274.
   Web of Science ®Google Scholar
• S.F. Yuan, A.P. Liao, and Y. Lei, Least squares Hermitian solution of the matrix equation (AXB,CXD)=(E,F) with the least norm over the skew field of quaternions, Math. Comput. Model. 48 (2008), pp. 91–100. doi: 10.1016/j.mcm.2007.08.009
   Web of Science ®Google Scholar
• S.F. Yuan, Q.W. Wang, and X. Zhang, Least-squares problem for the quaternion matrix equation AXB + CY D = E over different constrained matrices, Int. J. Comput. Math. 90 (2013), pp. 565–576. doi: 10.1080/00207160.2012.722626
   Web of Science ®Google Scholar
• S.F. Yuan, Q.W. Wang, and X. Duan, On solutions of the quaternion matrix equation AX = B and their applications in color image restoration, Appl. Math. Comput. 221 (2013), pp. 10–20. doi: 10.1016/j.amc.2013.05.069
   Web of Science ®Google Scholar
• Y. Zhang and R.H. Wang, The exact solution of a system of quaternion matrix equations involving η-Hermicity, Appl. Math. Comput. 222 (2013), pp. 201–209. doi: 10.1016/j.amc.2013.07.025
   Web of Science ®Google Scholar
• Y. Zhang, D. Jiang, and J. Wang, A recurrent neural network for solving Sylvester equation with time-varying coefficients, IEEE Trans. Neural Netw. 13 (2002), pp. 1053–1063. doi: 10.1109/TNN.2002.1031938
   PubMed Web of Science ®Google Scholar
• F.X. Zhang, W.S. Mu, Y. Li, and J.L. Zhao, Special least squares solutions of the quaternion matrix equation AXB + CXD = E, Comput. Math. Appl 72 (2016), pp. 1426–1435. doi: 10.1016/j.camwa.2016.07.019
   Web of Science ®Google Scholar
• F.X. Zhang, M.S. Wei, Y. Li, and J.L. Zhao, Special least squares solutions of the quaternion matrix equation AX = B with applications, Appl. Math. Comput 270 (2015), pp. 425–433. doi: 10.1016/j.amc.2015.08.046
   Web of Science ®Google Scholar
• F.X. Zhang, M.S. Wei, Y. Li, and J.L. Zhao, An efficient method for special least squares solution of the complex matrix equation (AXB,CXD)=(E,F), Comput. Math. Appl 76 (2018), pp. 2001–2010. doi: 10.1016/j.camwa.2018.07.044
   Web of Science ®Google Scholar
• F.X. Zhang, M.S. Wei, Y. Li, and J.L. Zhao, The minimal norm least squares Hermitian solution of the complex matrix equation AXB + CXD = E, J. Frankl. Inst 355 (2018), pp. 1296–1310. doi: 10.1016/j.jfranklin.2017.12.023
   Web of Science ®Google Scholar
• B. Zhou, Z.Y. Li, G.R. Duan, and Y. Wang, Weighted least squares solutions to general coupled Sylvester matrix equations, J. Comput. Appl. Math. 224 (2009), pp. 759–776. doi: 10.1016/j.cam.2008.06.014
   Web of Science ®Google Scholar