Parametric solutions to Sylvester-conjugate matrix equations

By two recently proposed operations with respect to complex matrices, a simple explicit solution to the Sylvester-conjugate matrix equation is given in a finite series form. The obtained solution can also be equivalently expressed in terms of the so-called controllability-like matrix and observability-like matrix. The proposed solution can provide all the degrees of freedom which is represented by a free parameter matrix. An illustrative example is employed to show the effectiveness of the proposed method.