Quaternion Matrix Equation AXB Research Articles

Dual Quaternion Matrix Equation AXB = C with Applications

Dual quaternions have wide applications in automatic differentiation, computer graphics, mechanics, and others. Due to its application in control theory, matrix equation AXB=C has been extensively studied. However, there is currently limited information on matrix equation AXB=C regarding the dual quaternion algebra. In this paper, we provide the necessary and sufficient conditions for the solvability of dual quaternion matrix equation AXB=C, and present the expression for the general solution when it is solvable. As an application, we derive the ϕ-Hermitian solutions for dual quaternion matrix equation AXAϕ=C, where the ϕ-Hermitian extends the concepts of Hermiticity and η-Hermiticity. Lastly, we present a numerical example to verify the main research results of this paper.

(R, S)-(Skew) Symmetric Solutions to Matrix Equation AXB = C over Quaternions

(R,S)-(skew) symmetric matrices have numerous applications in civil engineering, information theory, numerical analysis, etc. In this paper, we deal with the (R,S)-(skew) symmetric solutions to the quaternion matrix equation AXB=C. We use a real representation Aτ to obtain the necessary and sufficient conditions for AXB=C to have (R,S)-(skew) symmetric solutions and derive the solutions when it is consistent. We also derive the least-squares (R,S)-(skew) symmetric solution to the above matrix equation.

An efficient real representation method for least squares problem of the quaternion constrained matrix equation AXB + CY D = E

Let and represent the sets of all η-Hermitian quaternion matrices and η-anti-Hermitian quaternion matrices, respectively. On the basis of the real representation matrix of a quaternion matrix and its particular structure, we convert the least squares problem of the quaternion matrix equation AXB + CY D = E over into the corresponding problem of the real matrix equation over free variables, and then we establish its unique minimal norm least squares solution. Our resulting expressions are expressed only by real matrices, and the algorithm only includes real operations. Consequently, they are very simple and convenient. Compared with the existing method [S.F. Yuan, Q.W. Wang, and X. Zhang, Least-squares problem for the quaternion matrix equation AXB + CYD = E over different constrained matrices, Int. J. Comput. Math. 90 (2013), pp. 565–576], the final two examples show that our method is more efficient and superior.

Least-squares bihermitian and skew bihermitian solutions of the quaternion matrix equation AXB = C

In this paper, by using the special structure of the real representation of quaternion matrices, the properties of the Moore–Penrose inverse of quaternion matrices and the Kronecker product of matrices, we study least-squares bihermitian and skew bihermitian solutions of the quaternion matrix equation AXB = C, respectively. First we study the special structures of quaternion bihermitian and skew bihermitian matrices. Then the problem of solving the least-squares bihermitian and skew bihermitian solutions of the quaternion matrix equation AXB = C can be transformed into particular real linear system using these special structures. We deduce the form of all least-squares bihermitian and skew bihermitian solutions of the quaternion matrix equation AXB = C and propose real structure-preserving algorithms to compute the minimal norm least-squares bihermitian and skew bihermitian solution. All the computation process only involve real arithmetic. Numerical examples are provided to verify the effectiveness of our algorithms. Particularly, we also propose the conditions and computation method of bihermitian and skew bihermitian solutions of quaternion matrix equation AXB = C.

Determinantal Representations for the Solution of the Generalized Sylvester Quaternion Matrix Equation

By using the determinantal representations of the Moore–Penrose inverse matrix, within the framework of the theory of noncommutative column–row determinants, we obtain determinantal representations (analogs of the Cramer rule) for the solution of the generalized Sylvester quaternion matrix equation AXB + CYD = E .

An Iterative algorithm for $\eta$-(anti)-Hermitian least-squares solutions of quaternion matrix equations

Recently, some research has been devoted to finding the explicit forms of the η-Hermitian and η-anti-Hermitian solutions of several kinds of quaternion matrix equations and their associated least-squares problems in the literature. Although exploiting iterative algorithms is superior than utilizing the explicit forms in application, hitherto, an iterative approach has not been offered for finding η-(anti)-Hermitian solutions of quaternion matrix equations. The current paper deals with applying an efficient iterative manner for determining η-Hermitian and η-anti-Hermitian least-squares solutions corresponding to the quaternion matrix equation AXB + CY D = E. More precisely, first, this paper establishes some properties of the η-Hermitian and η-anti-Hermitian matrices. These properties allow for the demonstration of how the well-known conjugate gradient least- squares (CGLS) method can be developed for solving the mentioned problem over the η-Hermitian and η-anti-Hermitian matrices. In addition, the convergence properties of the proposed algorithm are discussed with details. In the circumstance that the coefficient matrices are ill-conditioned, it is suggested to use a preconditioner for accelerating the convergence behavior of the algorithm. Numerical experiments are reported to reveal the validity of the elaborated results and feasibility of the proposed iterative algorithm and its preconditioned version.

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

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.

Determinantal representations of the Drazin inverse over the quaternion skew field with applications to some matrix equations

Within the framework of the theory of the column and row determinants, we obtain determinantal representations of the Drazin inverse both for Hermitian and arbitrary matrices over the quaternion skew field. Using the obtained determinantal representations of the Drazin inverse we get explicit representation formulas (analogs of Cramer’s rule) for the Drazin inverse solutions of a quaternion matrix equation AXB=D and consequently AX=D, and XB=D in two cases if A,B are Hermitian or arbitrary.

An Efficient Algorithm for the Reflexive Solution of the Quaternion Matrix EquationAXB+CXHD=F

We propose an iterative algorithm for solving the reflexive solution of the quaternion matrix equationAXB+CXHD=F. When the matrix equation is consistent over reflexive matrixX, a reflexive solution can be obtained within finite iteration steps in the absence of roundoff errors. By the proposed iterative algorithm, the least Frobenius norm reflexive solution of the matrix equation can be derived when an appropriate initial iterative matrix is chosen. Furthermore, the optimal approximate reflexive solution to a given reflexive matrixX0can be derived by finding the least Frobenius norm reflexive solution of a new corresponding quaternion matrix equation. Finally, two numerical examples are given to illustrate the efficiency of the proposed methods.

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

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.

Extreme Ranks of Real Matrices in Solution of the Quaternion Matrix Equation AXB = C with Applications

In this paper, for a consistent quaternion matrix equation AXB = C, the formulas are established for maximal and minimal ranks of real matrices X1, X2, X3, X4 in solution X = X1 + X2i + X3j + X4k. A necessary and sufficient condition is given for the existence of a real solution of the quaternion matrix equation. The expression is also presented for the general solution to this equation when the solvability conditions are satisfied. Moreover, necessary and sufficient conditions are given for this matrix equation to have a complex solution or a pure imaginary solution. As applications, the maximal and minimal ranks of real matrices E, F, G, H in a generalized inverse (A +Bi + Cj + Dk)- = E + Fi + Gj + Hk of a quaternion matrix A + Bi + Cj + Dk are also considered. In addition, a necessary and sufficient condition is derived for the quaternion matrix equations A1XB1 = C1 and A2XB2 = C2 to have a common real solution.

Quaternion generalized singular value decomposition and its applications

This paper derives a theorem of generalized singular value decomposition of quaternion matrices (QGSVD), studies the solution of general quaternion matrix equation AXB −CYD=E, and obtains quaternionic Roth’s theorem. This paper also suggests sufficient and necessary conditions for the existence and uniqueness of solutions and explicit forms of the solutions of the equation.