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 Determinantal Representations of General and (Skew-)Hermitian Solutions to the Generalized Sylvester-Type Quaternion Matrix Equation

2019 ◽  
Vol 2019 ◽  
pp. 1-14 ◽  
Author(s):  
Ivan I. Kyrchei
Keyword(s):  
Matrix Equation ◽  
Quaternion Matrix ◽  
Sylvester Matrix Equation ◽  
Skew Field ◽  
Determinantal Representation ◽  
Sylvester Matrix ◽  
Generalized Sylvester Matrix Equation ◽  
Representation Formulas ◽  
Quaternion Matrix Equation

In this paper, we derive explicit determinantal representation formulas of general, Hermitian, and skew-Hermitian solutions to the generalized Sylvester matrix equation involving ⁎-Hermicity AXA⁎+BYB⁎=C over the quaternion skew field within the framework of the theory of noncommutative column-row determinants.

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>

scholarly journals Classification and approximation of solutions to Sylvester matrix equation

Filomat ◽  
2019 ◽  
Vol 33 (13) ◽  
pp. 4261-4280 ◽  
Author(s):  
Bogdan Djordjevic ◽  
Nebojsa Dincic
Keyword(s):  
Least Squares ◽  
Matrix Equation ◽  
Infinitely Many Solutions ◽  
Sylvester Matrix Equation ◽  
Minimal Norm ◽  
Sylvester Matrix

In this paperwesolve Sylvester matrix equation with infinitely-many solutions and conduct their classification. If the conditions for their existence are not met, we provide a way for their approximation by least-squares minimal-norm method.

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 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.

scholarly journals The least squares η-Hermitian problems of quaternion matrix equation AHXA + BHYB=C

Filomat ◽  
2014 ◽  
Vol 28 (6) ◽  
pp. 1153-1165 ◽  
Author(s):  
Shi-Fang Yuan ◽  
Qing-Wen Wang ◽  
Zhi-Ping Xiong
Keyword(s):  
Least Squares ◽  
Matrix Equation ◽  
Kronecker Product ◽  
Generalized Inverse ◽  
Hermitian Matrix ◽  
Quaternion Matrix ◽  
Hermitian Matrices ◽  
Quaternion Matrices ◽  
Quaternion Matrix Equation ◽  
Product Of Matrices

For any A=A1+A2j?Qnxn and ?? {i,j,k} denote A?H = -?AH?. If A?H = A,A is called an ?-Hermitian matrix. If A?H =-A,A is called an ?-anti-Hermitian matrix. Denote ?-Hermitian matrices and ?-anti-Hermitian matrices by ?HQnxn and ?AQnxn, respectively. In this paper, we consider the least squares ?-Hermitian problems of quaternion matrix equation AHXA+ BHYB = C by using the complex representation of quaternion matrices, the Moore-Penrose generalized inverse and the Kronecker product of matrices. We derive the expressions of the least squares solution with the least norm of quaternion matrix equation AHXA + BHYB = C over [X,Y] ? ?HQnxn x ?HQkxk, [X,Y] ? ?AQnxn x ?AQkxk, and [X,Y] ? ?HQnxn x ?AQkxk, respectively.

Least Squares Skew Bisymmetric Solution for a Kind of Quaternion Matrix Equation

2011 ◽  
Vol 50-51 ◽  
pp. 190-194 ◽  
Author(s):  
Shi Fang Yuan ◽  
Han Dong Cao
Keyword(s):  
Least Squares ◽  
Matrix Equation ◽  
Kronecker Product ◽  
Special Structure ◽  
Complex Representation ◽  
Quaternion Matrix ◽  
Quaternion Matrices ◽  
Quaternion Matrix Equation ◽  
Product Of Matrices

In this paper, by using the Kronecker product of matrices and the complex representation of quaternion matrices, we discuss the special structure of quaternion skew bisymmetric matrices, and derive the expression of the least squares skew bisymmetric solution of the quaternion matrix equation AXB =C with the least norm.

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.

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