Special arrowhead least squares solution of the quaternion generalized Sylvester matrix equation

In this paper, we study special arrowhead solution of the quaternion generalized Sylvester matrix equation by using semi-tensor product of matrices. We first propose a real vector representation of quaternion matrix and study its properties. By means of this real vector representation, semi-tensor product of matrices and Moore-Penrose inverse, we solve the least squares arrowhead Hermitian (skew-Hermitian) problem of the quaternion generalized Sylvester matrix equation. The expressions of these solutions only involve real arithmetic, which makes the procedure computation more flexible and efficient.