Search

In this paper, a new real representation of reduced biquaternion matrix is proposed, and the solutions of the reduced biquaternion matrix equations XF−AX=BY$$ XF- AX= BY $$ and XF−AX˜=BY$$ XF-A\tilde{X}= BY $$ are solved by means of this method. The corresponding numerical algorithm is provided, and the effectiveness of this method is verified by numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we put forward two efficient methods for solving the commutative quaternion Stein matrix equation X + A X B = C . By combining real representation of a commutative quaternion matrix and H -representation or generalized H -representation of special matrices, we investigate the minimal norm least squares L-structured and generalized L-structured solutions of the previous commutative quaternion matrix equation and derive their expressions. In this way, we first present a theoretical study on extending L-structured real matrices to generalized L-structured matrices, and introduce some generalized L-structured matrices. Based on them, we then discuss their applications in commutative quaternion Stein matrix equation. The algorithms only involve real operations. Consequently, it is very simple and convenient, and it can be used for all kinds of commutative quaternion matrix equation with similar problems. Furthermore, an illustrative example is provided to show the feasibility of the given methods. [ABSTRACT FROM AUTHOR]

In this paper, we study the bisymmetric and skew bisymmetric solutions of quaternion generalized Lyapunov equation. With the help of semi-tensor product of matrices, some new conclusions on the expansion rules of row and column of matrix product on quaternion matrices are proposed and applied to the calculation of quaternion matrix equation. Using the H-representation method, the independent elements are extracted according to the structural characteristics of bisymmetric matrix and skew bisymmetric matrix, so as to simplify the operation process. Finally, it is compared with the real vector representation method of quaternion matrix equation to illustrate the effectiveness and superiority of the proposed method. [ABSTRACT FROM AUTHOR]

We propose a practical method for computing the singular value decomposition of dual quaternion matrices. The dual quaternion Householder matrix is first proposed, and by combining the properties of dual quaternions, we can implement the transformation of a dual quaternion matrix to a bidiagonalized dual number matrix. We have proven that the singular values of a dual quaternion matrix are same to the singular values of its bidiagonalized dual number matrix. Numerical experiment demonstrates the effectiveness of our proposed method for computing the singular value decomposition of dual quaternion matrices. [ABSTRACT FROM AUTHOR]

In this paper, using the real representation method, we study the reduced biquaternion matrix equation A X = B . Taking advantage of the special structure of real representation of reduced biquaternion, we transform the problem of reduced biquaternion matrix into corresponding problem of real matrix. We propose the expressions of the special minimal norm least squares solution of the reduced biquaternion matrix equation A X = B , and the corresponding algorithms only perform real arithmetic. Numerical examples are provided to illustrate that our algorithm are efficient and easily understood. We also apply the minimal norm pure imaginary least squares reduced biquaternion solution to color image restoration. [ABSTRACT FROM AUTHOR]