A new method based on semi-tensor product of matrices for solving Generalized Lyapunov Equation on the quaternion skew-filed

In this paper, we propose a new method based on semi-tensor product of matrices for solving quaternion Generalized Lyapunov Equation ${A^H}X + XA + \sum\limits_{j = 1}^m {N_j^H} X{N_j} + {C^H}C = 0$. First, we define a real vector representation of quaternion matrix and study its properties. By using semi-tensor product of matrices and this real vector representation, we transform the problem of quaternion matrix equation into the corresponding real matrix equation. And then, by studying the relationship between the real vector representations of Hermitian matrix and their independent elements, we reduce the complexity of the original problem. We derive the expression of the least squares Hermitian solution for the quaternion Generalized Lyapunov Equation ${A^H}X + XA + \sum\limits_{j = 1}^m {N_j^H} X{N_j} + {C^H}C = 0$. All the expressions of solutions only involve real arithmetic. We propose the corresponding algorithms and give numerical experiments to illustrate the validity of our methods.