The reflexive re-nonnegative definite solution to a quaternion matrix equation

In this paper a necessary and sufficient condition is established for the existence of the reflexive re-nonnegative definite solution to the quaternion matrix equation AX A∗ = B, where ∗ stands for conjugate transpose. The expression of such solution to the matrix equation is also given. Furthermore, a necessary and sufficient condition is derived for the existence of the general re-nonnegative definite solution to the quaternion matrix equation A1X1A∗1 + A2X2A∗2 = B. The representation of such solution to the matrix equation is given.