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

Let be the set of all matrices over the real quaternion algebra. , , , where is the conjugate of the quaternion . We call that is -Hermitian, if , ; is -bihermitian, if . We in this paper, present the solvability conditions and the general -Hermitian solution to a system of linear real quaternion matrix equations. As an application, we give the necessary and sufficient conditions for the systemto have an -bihermitian solution. We establish an expression of the -bihermitian to the system when it is solvable. We also obtain a criterion for a quaternion matrix to be -bihermitian. Moreover, we provide an algorithm and a numerical example to illustrate the theory developed in this paper.

[1]  Qingxiang Xu.  Common hermitian and positive solutions to the adjointable operator equations AX = C, XB = D , 2008 .

[2]  Shifang Yuan,et al.  Two special kinds of least squares solutions for the quaternion matrix equation AXB+CXD=E , 2012 .

[3]  J. K. Baksalary,et al.  The matrix equation AXB+CYD=E , 1980 .

[4]  Fuzhen Zhang.  Quaternions and matrices of quaternions , 1997 .

[5]  Yuan Lei,et al.  Least squares Hermitian solution of the matrix equation (AXB, CXD) = (E, F) with the least norm over the skew field of quaternions , 2008, Math. Comput. Model..

[6]  Roger A. Horn,et al.  A generalization of the complex Autonne–Takagi factorization to quaternion matrices , 2012 .

[7]  Masoud Hajarian,et al.  A lower bound for the product of eigenvalues of solutions to matrix equations , 2009, Appl. Math. Lett..

[8]  Qingwen Wang,et al.  A real quaternion matrix equation with applications , 2013 .

[9]  Soo-Chang Pei,et al.  Efficient implementation of quaternion Fourier transform, convolution, and correlation by 2-D complex FFT , 2001, IEEE Trans. Signal Process..

[10]  Mehdi Dehghan,et al.  Computational methods for solving fully fuzzy linear systems , 2006, Appl. Math. Comput..

[11]  Qing-Wen Wang,et al.  Bisymmetric and centrosymmetric solutions to systems of real quaternion matrix equations , 2005 .

[12]  N. K. Thakare,et al.  Generalized inverses and operator equations , 1979 .

[13]  Danilo P. Mandic,et al.  On the unitary diagonalisation of a special class of quaternion matrices , 2011, Appl. Math. Lett..

[14]  Zhong-Zhi Bai,et al.  The Constrained Solutions of Two Matrix Equations , 2002 .

[15]  Mohammad Sal Moslehian,et al.  On the Hermitian solutions to a system of adjointable operator equations , 2012 .

[16]  Yongge Tian,et al.  A simultaneous decomposition of a matrix triplet with applications , 2011, Numer. Linear Algebra Appl..

[17]  Huang Liping,et al.  The matrix equation AXB+CYD=Eover a simple artinian ring , 1995 .

[18]  Guiping Xu,et al.  ON SOLUTIONS OF MATRIX EQUATION AXB + CYD = F , 1998 .

[19]  Xi-Yan Hu,et al.  The solvability conditions for the inverse eigenproblems of symmetric and generalized centro-symmetric matrices and their approximations☆ , 2006 .

[20]  Yoshio Takane,et al.  On consistency, natural restrictions and estimability under classical and extended growth curve models , 2009 .

[21]  J. Woude,et al.  A system of real quaternion matrix equations with applications , 2009 .

[22]  Mehdi Dehghan,et al.  Finite iterative algorithms for the reflexive and anti-reflexive solutions of the matrix equation A1X1B1+A2X2B2=C , 2009, Math. Comput. Model..

[23]  Sujit Kumar Mitra,et al.  Hermitian and Nonnegative Definite Solutions of Linear Matrix Equations , 1976 .

[24]  Danilo P. Mandic,et al.  The Quaternion LMS Algorithm for Adaptive Filtering of Hypercomplex Processes , 2009, IEEE Transactions on Signal Processing.

[26]  J. Woude,et al.  Almost non-interacting control by measurement feedback , 1987 .

[27]  S. Leo,et al.  Right eigenvalue equation in quaternionic quantum mechanics , 2000, math-ph/0002051.

[28]  Qing-Wen Wang,et al.  Ranks and the least-norm of the general solution to a system of quaternion matrix equations , 2009 .

[29]  Yanping Sheng,et al.  The least-squares solutions of inconsistent matrix equation over symmetric and antipersymmetric matrices , 2003, Appl. Math. Lett..

[30]  Yang Zhang,et al.  The exact solution of a system of quaternion matrix equations involving η-Hermicity , 2013, Appl. Math. Comput..

[31]  Fuzhen Zhang.  Geršgorin type theorems for quaternionic matrices , 2007 .

[32]  Salvatore D. Morgera,et al.  On the reducibility of centrosymmetric matrices — Applications in engineering problems , 1989 .

[33]  Qing-Wen Wang,et al.  Consistency for bi(skew)symmetric solutions to systems of generalized Sylvester equations over a finite central algebra , 2002 .

[35]  Danilo P. Mandic,et al.  Augmented second-order statistics of quaternion random signals , 2011, Signal Process..

[36]  The common positive solution to adjointable operator equations with an application , 2012 .

[37]  A. Liao,et al.  On Hermitian positive definite solution of the matrix equation X - Σ m i=1 A i * X r A i = Q , 2009 .

[38]  A. B. Ögüler.  The equation AXB+CYD = E over a principal ideal domain , 1991 .

[39]  G. Styan,et al.  Equalities and Inequalities for Ranks of Matrices , 1974 .

[40]  Zhuo-Heng He,et al.  Some matrix equations with applications† , 2012 .

[41]  A. Dajić,et al.  Equations ax = c and xb = d in rings and rings with involution with applications to Hilbert space operators , 2008 .

[42]  A. Liao,et al.  On the existence of Hermitian positive definite solutions of the matrix equation Xs + A*X-tA = Q , 2008 .

[43]  Nicolas Le Bihan,et al.  Singular value decomposition of quaternion matrices: a new tool for vector-sensor signal processing , 2004, Signal Process..

[44]  M. Dehghan,et al.  The general coupled matrix equations over generalized bisymmetric matrices , 2010 .

[45]  Danilo P. Mandic,et al.  Quaternion-Valued Stochastic Gradient-Based Adaptive IIR Filtering , 2010, IEEE Transactions on Signal Processing.

[46]  A. Cantoni,et al.  Eigenvalues and eigenvectors of symmetric centrosymmetric matrices , 1976 .

[47]  R. D. Hill,et al.  On perhermitian matrices , 1990 .

[48]  Stephen J. Sangwine,et al.  Colour image edge detector based on quaternion convolution , 1998 .

[49]  Xin-Guo Liu,et al.  On the positive definite solutions of the matrix equations Xs±ATX−tA=In☆ , 2003 .

[50]  Nicolas Le Bihan,et al.  Quaternion-MUSIC for vector-sensor array processing , 2006, IEEE Transactions on Signal Processing.