On Hermitian Solutions of the Split Quaternion Matrix Equation $$AXB+CXD=E$$ A X B + C X D = E

2017 ◽  
Vol 27 (4) ◽  
pp. 3235-3252 ◽  
Author(s):  
Shi-Fang Yuan ◽  
Qing-Wen Wang ◽  
Yi-Bin Yu ◽  
Yong Tian
Keyword(s):  
Matrix Equation ◽  
Quaternion Matrix ◽  
Split Quaternion ◽  
Quaternion Matrix Equation ◽  
Split Quaternion Matrix

scholarly journals A Real Representation Method for Solving Yakubovich-j-Conjugate Quaternion Matrix Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Caiqin Song ◽  
Jun-e Feng ◽  
Xiaodong Wang ◽  
Jianli Zhao
Keyword(s):  
Matrix Equation ◽  
Equivalent Form ◽  
Closed Form Solution ◽  
Coefficient Matrix ◽  
Form Solution ◽  
Complex Conjugate ◽  
Quaternion Matrix ◽  
Real Representation ◽  
Quaternion Matrix Equation ◽  
Representation Method

A new approach is presented for obtaining the solutions to Yakubovich-j-conjugate quaternion matrix equationX−AX^B=CYbased on the real representation of a quaternion matrix. Compared to the existing results, there are no requirements on the coefficient matrixA. The closed form solution is established and the equivalent form of solution is given for this Yakubovich-j-conjugate quaternion matrix equation. Moreover, the existence of solution to complex conjugate matrix equationX−AX¯B=CYis also characterized and the solution is derived in an explicit form by means of real representation of a complex matrix. Actually, Yakubovich-conjugate matrix equation over complex field is a special case of Yakubovich-j-conjugate quaternion matrix equationX−AX^B=CY. Numerical example shows the effectiveness of the proposed results.

The Solution Set to the Quaternion Matrix Equation $AX-\overline{X}B=0$

2012 ◽  
Vol 19 (01) ◽  
pp. 175-180 ◽  
Author(s):  
Lianggui Feng ◽  
Wei Cheng
Keyword(s):  
Matrix Equation ◽  
Solution Set ◽  
Quaternion Matrix ◽  
Quaternion Matrix Equation ◽  
Clear Description

We give a clear description of the solution set to the quaternion matrix equation [Formula: see text], where A, B are known, X is unknown and [Formula: see text] denotes the usual conjugate of X.

scholarly journals Least Squares Pure Imaginary Solution and Real Solution of the Quaternion Matrix EquationAXB+CXD=Ewith the Least Norm

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Shi-Fang Yuan
Keyword(s):  
Least Squares ◽  
Matrix Equation ◽  
Kronecker Product ◽  
Generalized Inverse ◽  
Complex Representation ◽  
Real Solution ◽  
Quaternion Matrix ◽  
Quaternion Matrices ◽  
Quaternion Matrix Equation ◽  
Product Of Matrices

Using the Kronecker product of matrices, the Moore-Penrose generalized inverse, and the complex representation of quaternion matrices, we derive the expressions of least squares solution with the least norm, least squares pure imaginary solution with the least norm, and least squares real solution with the least norm of the quaternion matrix equationAXB+CXD=E, respectively.

Least-norm and Extremal Ranks of the Least Square Solution to the Quaternion Matrix Equation AXB=C Subject to Two Equations

2014 ◽  
Vol 21 (03) ◽  
pp. 449-460 ◽  
Author(s):  
Yubao Bao
Keyword(s):  
Matrix Equation ◽  
Least Square ◽  
Matrix Equations ◽  
Quaternion Matrix ◽  
Consistent System ◽  
Quaternion Matrix Equation

In this paper, we give the expression of the least square solution of the linear quaternion matrix equation AXB=C subject to a consistent system of quaternion matrix equations D1X=F1, XE2=F2, and derive the maximal and minimal ranks and the least-norm of the above mentioned solution. The finding of this paper extends some known results in the literature.

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