In this paper, we study the Hankel and Toeplitz solutions of reduced biquaternion matrix equation (1.1). Using semi-tensor product of matrices, the reduced biquaternion matrix equation (1.1) can be transformed into a general matrix equation of the form $AX=B$. Then, due to the special structure of Hankel matrix and Toeplitz matrix, the independent elements of Hankel matrix or Toeplitz matrix can be extracted by combing the $\mathcal{H}$-representation method of matrix, so as to reduce the elements involved in the operation in the process of solving matrix equation and reduce the complexity of the problem. Finally, by using Moore-Penrose generalized inverse, the necessary and sufficient conditions for the existence of solutions of reduced biquaternion matrix equation (1.1) are given, and the corresponding numerical examples are given.

In this paper, we study several constrainted least squares solutions of quaternion Sylvester matrix equation. We first propose a real vector representation of quaternion matrix and study its properties. By using this real vector representation, semi-tensor product of matrices, swap matrix and Moore-Penrose inverse, we derive compatible conditions and the expressions of several constrainted least squares solutions of quaternion Sylvester equation.

In this paper, we investigate the mixed solution of reduced biquaternion matrix equation $\sum _{i=1}^n{A}_i{X}_i{B}_i=E$ with sub-matrix constraints. With the help of ${\mathcal{L}}_{\mathcal{C}}$-representation and the properties of vector operator based on semi-tensor product of reduced biquaternion matrices, the reduced biquaternion matrix equation (1.1) can be transformed into linear equations. A systematic method, $\mathcal{G}\mathcal{H}$-representation, is proposed to decrease the number of variables of a special unknown reduced biquaternion matrix and applied to solve the least squares problem of linear equations. Meanwhile, we give the necessary and sufficient conditions for the compatibility of reduced biquaternion matrix equation (1.1) under sub-matrix constraints. Numerical examples are given to demonstrate the results. The method proposed in this paper is applied to color image restoration.

In this paper, we propose a real vector representation of reduced quaternion matrix and study its properties. By using this real vector representation, Moore-Penrose inverse, and semi-tensor product of matrices, we study some kinds of solutions of reduced biquaternion matrix equation (1.1). Several numerical examples show that the proposed algorithm is feasible at last.