<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>

In this paper, we consider the split quaternion matrix equation X−Af(X)B=C, f(X)∈{X,XH,XiH,XjHXkH}. The H representation method has the characteristics of transforming a matrix with a special structure into a column vector with independent elements. By using the real representation of split quaternion matrices, H representation method, the Kronecker product of matrices and the Moore-Penrose generalized inverse, we convert the split quaternion matrix equation into the real matrix equation, and derive the sufficient and necessary conditions and the general solution expressions for the (skew) bisymmetric solution of the original equation. Moreover, we provide numerical algorithms and illustrate the efficiency of our method by two numerical examples.

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In this paper, we investigate the least squares solution with the minimal norm to the system (1.1) over reduced biquaternion via complex representation of reduced biquaternion tensors and the Moore-Penrose inverse of tensors. Besides, we establish some necessary and sufficient conditions for the solvability to the above system and give an expression of the general solution to the system when the solvability conditions are met. Moreover, the algorithm and numerical example are presented to verify the main results of this paper.

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 and XF − A 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.