The η-bihermitian solution to a system of real quaternion matrix equations

References

• Took CC, Mandic DP, Zhang FZ. On the unitary diagonalization of a special class of quaternion matrices. Appl. Math. Lett. 2011;24:1806–1809.
   Web of Science ®Google Scholar
• Horn RA, Zhang F. A generalization of the complex Autonne-Takagi factorization to quaternion matrices. Linear Multilinear Algebra. 2012;60:1239–1244.
   Web of Science ®Google Scholar
• Wang QW, Sun JH, Li SZ. Consistency for bi(skew)symmetric solutions to systems of generalized Sylvester equations over a finite central algebra. Linear Algebra Appl. 2002;353:169–182.
   Web of Science ®Google Scholar
• Zhang FZ. Quaternions and matrices of quaternions. Linear Algebra Appl. 1997;251:21–57.
   Web of Science ®Google Scholar
• Bihan NL, Mars J. Singular value decomposition of quaternion matrices: a new tool for vector-sensor signal processing. Sign. Process. 2004;84:1177–1199.
   Web of Science ®Google Scholar
• Leo SD. Right eigenvalue equation in quaternionic quantum mechanics. J. Phys. A. 2000;33: 2971–2995.
   Google Scholar
• Miron S, Bihan NL, Mars J. Quaternion-music for vector-sensor array processing. IEEE Trans. Sign. Process. 2009;54:1218–1229.
   Web of Science ®Google Scholar
• Pei SC, Ding JJ, Chang JH. Efficient implementation of quaternion Fourier transform, convolution, and correlation by 2-D complex FFT. IEEE Trans. Sign. Process. 2001;49:2783–2797.
   Web of Science ®Google Scholar
• Sangwine SJ. Colour image edge detector based on quaternion convolution. Electron. Lett. 1998;34:969–971.
   Web of Science ®Google Scholar
• Zhang FZ. Gersgorin type theorems for quaternionic matrices. Linear Algebra Appl. 2007;424:139–53.
   Web of Science ®Google Scholar
• Dehghan M, Hajarian M. Finite iterative algorithms for the reflexive and anti-reflexive solutions of the matrix equation A1X B1 + A2X B2 = C. Math. Comput. Model. 2009;49:1937–1959.
   Web of Science ®Google Scholar
• Dehghan M, Hajarian M. A lower bound for the product of eigenvalues of solutions to matrix equations. Appl. Math Lett. 2009;22:1786–1788.
   Web of Science ®Google Scholar
• Dehghan M, Hajarian M. The general coupled matrix equations over generalized bisymmetric matrices. Linear Algebra Appl. 2010;432:1531–1552.
   Web of Science ®Google Scholar
• Dehghan M, Hashemi B, Ghatee M. Computational methods for solving fully fuzzy linear systems. Appl. Math. Comput. 2006;179:328–343.
   Web of Science ®Google Scholar
• Duan XF, Liao AP. On the existence of Hermitian positive definite solutions of the matrix equation Xs + A*X–t A = Q. Linear Algebra Appl. 2008;429:673–687.
   Web of Science ®Google Scholar
• Duan XF, Liao AP. On Hermitian positive definite solution of the matrix equation X –. J. Comput. Appl. Math. 2009;229:27–36.
   Web of Science ®Google Scholar
• Farid FO, Moslehian MS, Wang QW, Wu ZC. On the Hermitian solutions to a system of adjointable operator equations. Linear Algebra Appl. 2012;437:1854–1891.
   Web of Science ®Google Scholar
• He ZH, Wang QW. A real quaternion matrix equation with applications. Linear Multilinear Algebra. 2013;61:725–740.
   Web of Science ®Google Scholar
• Hell RD, Bates RG, Waters SR. On perhermitian matrices. SIAM J. Matrix Anal. Appl. 1990;11:173–179.
   Web of Science ®Google Scholar
• Huang LP, Zeng Q. The solvability of matrix equation AXB + CYD = E over a simple Arinian ring. Linear Multilinear Algebra. 1995;38:225–232.
   Google Scholar
• Khatri CG, Mitra SK. Hermitian and nonnegative definite solutions of linear matrix equations. SIAM J. Appl. Math. 1976;31:579–585.
   Web of Science ®Google Scholar
• van der Woude JW. Almost noninteracting control by measurement feedback. Syst Control Lett. 1987;9:7–16.
   Web of Science ®Google Scholar
• Wang QW, van der Woude JW, Chang HX. A system of real quaternion matrix equations with applications. Linear Algebra Appl. 2009;431:2291–2303.
   Web of Science ®Google Scholar
• Wang QW, He ZH. Some matrix equations with applications. Linear Multilinear Algebra. 2012;60:1327–1353.
   Web of Science ®Google Scholar
• Xie DX, Hu XY, Sheng YP. The solvability conditions for the inverse eigenproblems of symmetric and generalized centro-symmetric matrices and their approximations. Linear Algebra Appl. 2006;418:142–152.
   Web of Science ®Google Scholar
• Xie DX, Sheng YP, Hu XY. The least-squares solutions of inconsistent matrix equation over symmetric and antipersymmetric matrices. Appl. Math. Lett. 2003;16:589–598.
   Web of Science ®Google Scholar
• Xie DX, Liao AP. The least-squares solution of inverse problems for antisymmetric and persymmetric matrices. J. Numer. Methods Comput. Appl. 2003;24:304–313.
   Google Scholar
• Yuan SF, Liao AP, Lei Y. 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. 2008;48:91–100.
   Web of Science ®Google Scholar
• Tian YG, Takane Y. On consistency, natural restrictions and estimability under classical and extended growth curve models. J. Stat. Plann. Inference. 2009;139:2445–2458.
   Web of Science ®Google Scholar
• Baksalary JK, Kala R. The matrix equation AXB + CYD = E. Linear Algebra Appl. 1980;30:141–147.
   Web of Science ®Google Scholar
• Özgüler AB. The matrix equation AXB + CYD = E over a principal ideal domain. SIAM J. Matrix Anal. Appl. 1991;12: 581–591.
   Web of Science ®Google Scholar
• Wang QW. The decomposition of pairwise matrices equations over an arbitrary skew field. Acta Math. Sinica. 1996;39:396–403.
   Google Scholar
• Phadke SV, Thakare NK. Generalized inverses and operator equations. Linear Algebra Appl. 1979;23:191–199.
   Web of Science ®Google Scholar
• Dajić A, Koliha JJ. Equations ax = c and xb = d in rings and rings with involution with applications to Hilbert space operator. Linear Algebra Appl. 2008;429: 1779–1809.
   Web of Science ®Google Scholar
• Xu QX. Common Hermitian and positive solutions to the adjointable operator equations AX = C, XB = D. Linear Algebra Appl. 2008;429:1–11.
   Web of Science ®Google Scholar
• Liao AP, Bai ZZ. The constrained solutions of two matrix equations. Acta Math. Sin. (Engl.Ser). 2002;18: 671–678.
   Web of Science ®Google Scholar
• Xu GP, Wei MS, Zheng DS. On solution of matrix equation AXB + CYD = F. Linear Algebra Appl. 1998;279:93–109.
   Web of Science ®Google Scholar
• Deng YB, Hu XY. On solutions of matrix equation AXAT + BYBT = C. J. Comput. Math. 2005;23:17–26.
   Web of Science ®Google Scholar
• Liu YH, Tian YG. A simultaneous decomposition of a matrix triplet with applications. Numer. Linear Algebra Appl. 2011;18:69–85.
   Web of Science ®Google Scholar
• Dong CZ, Wang QW, Zhang YP. The common positive solution to adjointable operators equations with an application. J. Math. Anal. Appl. 2012;396:670–679.
   Web of Science ®Google Scholar
• Took CC, Mandic DP. Augmented second-order statistics of quaternion random signals. Sign. Process. 2011;91:214–224.
   Web of Science ®Google Scholar
• Took CC, Mandic DP. The quaternion LMS algorithm for adaptive filtering of hypercomplex real world processes. IEEE Trans. Sign. Process. 2009;57:1316–1327.
   Web of Science ®Google Scholar
• Took CC, Mandic DP. Quaternion-valued stochastic gradient-based adaptive IIR filtering. IEEE Trans. Sign. Process. 2010;58:3895–3901.
   Web of Science ®Google Scholar
• Yuan SF, Wang QW. Two special kinds of least squares solutions for the quaternion matrix equation AXB + CXD = E. Electron. J. Linear Algebra. 2012;23:257–274.
   Web of Science ®Google Scholar
• Cantoni A, Butler P. Eigenvalues and eigenvectors of symmetric centrosymmetric matrices. Linear Algebra Appl. 1976;13:275–288.
   Web of Science ®Google Scholar
• Datta L, Morgers SD. On the reducibility of centrosymmetric matrices-applications in engineering problems. Circuits Syst. Sign. Proc. 1989;8:71–96.
   Web of Science ®Google Scholar
• Wang QW. Bisymmetric and centrosymmetric solutions to system of real quaternion matrix equations. Comput. Math. Appl. 2005;49:641–650.
   Web of Science ®Google Scholar
• Wang QW, Li CK. Ranks and the least-norm of the general solution to a system of quaternion matrix equations. Linear Algebra Appl. 2009;430:1626–1640.
   Web of Science ®Google Scholar
• Marsaglia G, Styan GPH. Equalities and inequalities for ranks of matrices. Linear Multilinear Algebra. 1974;2:269–292.
   Google Scholar
• Zhang Y, Wang RH. The exact solution of a system of quaternion matrix equations involving η-Hermicity. Appl. Math. Comput. 2013;222:201–209.
   Web of Science ®Google Scholar