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This paper mainly studies the least-squares solutions of matrix equation $ AXB = C $ under a semi-tensor product. According to the definition of the semi-tensor product, the equation is transformed into an ordinary matrix equation. Then, the least-squares solutions of matrix-vector and matrix equations respectively investigated by applying the derivation of matrix operations. Finally, the specific form of the least-squares solutions is given.

In this article, our interest is the quaternion matrix equation (AXB,DXE)=(C,F), and we study its minimal norm centrohermitian least squares solution and skew centrohermitian least squares solution. By applying of the real representation matrices of quaternion matrices and relative properties, we convert the quaternion least squares problems with constrained variables into the corresponding real least squares problems with free variables, and then we obtain the solutions of corresponding problems. The final results can be expressed only by real matrices and vectors, and thus the corresponding algorithms only involves real operations and avoid complex quaternion operations. Therefore, they are portable and convenient. In the end, we give two examples to verify the effectiveness of the purposed algorithms.

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 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 H-representation to solve quaternion matrix equation with special structure, which not only broadens the application scope of 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.

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.

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.