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.

scholarly journals Three special kinds of least squares solutions for the quaternion generalized Sylvester matrix equation

2021 ◽  
Vol 7 (4) ◽  
pp. 5029-5048
Author(s):  
Anli Wei ◽  
◽  
Ying Li ◽  
Wenxv Ding ◽  
Jianli Zhao ◽  
...  
Keyword(s):  
Least Squares ◽  
Matrix Equation ◽  
Quaternion Matrix ◽  
Sylvester Matrix Equation ◽  
Real Representation ◽  
Minimal Norm ◽  
Generalized Sylvester Matrix Equation ◽  
Quaternion Matrix Equation ◽  
Special Solutions ◽  
Application Scope

<abstract><p>In this paper, we propose an efficient method for some special solutions of the quaternion matrix equation $ AXB+CYD = E $. By integrating real representation of a quaternion matrix with $ \mathcal{H} $-representation, we investigate the minimal norm least squares solution of the previous quaternion matrix equation over different constrained matrices and obtain their expressions. In this way, we first apply $ \mathcal{H} $-representation to solve quaternion matrix equation with special structure, which not only broadens the application scope of $ \mathcal{H} $-representation, but further expands the research idea of solving quaternion matrix equation. The algorithms only include real operations. Consequently, it is very simple and convenient, and it can be applied to all kinds of quaternion matrix equation with similar problems. The numerical example is provided to illustrate the feasibility of our algorithms.</p></abstract>

scholarly journals Real Representation Approach to Quaternion Matrix Equation Involving ϕ-Hermicity

2019 ◽  
Vol 2019 ◽  
pp. 1-8 ◽  
Author(s):  
Xin Liu ◽  
Huajun Huang ◽  
Zhuo-Heng He
Keyword(s):  
Matrix Equation ◽  
Sufficient Conditions ◽  
Complete Characterization ◽  
Quaternion Matrix ◽  
Real Representation ◽  
Quaternion Matrix Equation ◽  
The Matrix ◽  
Hermitian Solution ◽  
Necessary And Sufficient

For a quaternion matrix A, we denote by Aϕ the matrix obtained by applying ϕ entrywise to the transposed matrix AT, where ϕ is a nonstandard involution of quaternions. A is said to be ϕ-Hermitian or ϕ-skew-Hermitian if A=Aϕ or A=−Aϕ, respectively. In this paper, we give a complete characterization of the nonstandard involutions ϕ of quaternions and their conjugacy properties; then we establish a new real representation of a quaternion matrix. Based on this, we derive some necessary and sufficient conditions for the existence of a ϕ-Hermitian solution or ϕ-skew-Hermitian solution to the quaternion matrix equation AX=B. Moreover, we give solutions of the quaternion equation when it is solvable.

scholarly journals The least squares Bisymmetric solution of quaternion matrix equation $ AXB = C $

2021 ◽  
Vol 6 (12) ◽  
pp. 13247-13257
Author(s):  
Dong Wang ◽  
◽  
Ying Li ◽  
Wenxv Ding
Keyword(s):  
Least Squares ◽  
Matrix Equation ◽  
Quaternion Matrix ◽  
Real Representation ◽  
Numerical Examples ◽  
Least Squares Problem ◽  
The Real ◽  
Quaternion Matrix Equation ◽  
Four Blocks ◽  
The Relationship

<abstract><p>In this paper, the idea of partitioning is used to solve quaternion least squares problem, we divide the quaternion Bisymmetric matrix into four blocks and study the relationship between the block matrices. Applying this relation, the real representation of quaternion, and M-P inverse, we obtain the least squares Bisymmetric solution of quaternion matrix equation $ AXB = C $ and its compatable conditions. Finally, we verify the effectiveness of the method through numerical examples.</p></abstract>

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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