Explicit solutions to the quaternion matrix equations X - AXF = C and X - A(X)over-tildeF = C

In the present paper, we investigate the quaternion matrix equation X - AXF = C and X - A (X) over tildeF = C. For convenience, we named the quaternion matrix equations X - AXF = C and X -A (X) over tildeF = C as quaternion Stein matrix equation and quaternion Stein-conjugate matrix equation. Based on the Kronecker map and complex representation of a quaternion matrix, we give the solution expressions of the quaternion Stein matrix equation and quaternion Stein-conjugate matrix equation. Through these expressions, we can easily obtain the solution of the above two equations. In order to compare the direct algorithm with the indirect algorithm, we propose an example to illustrate the effectiveness of the proposed method.