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We investigate the matrix equationX  −  A   X  ¯   B  =  C    . For convenience, the matrix equationX  −  A   X  ¯   B  =  Cis named as Kalman-Yakubovich-conjugate matrix equation. The explicit solution is constructed when the above matrix equation has unique solution. And this solution is stated as a polynomial of coefficient matrices of the matrix equation. Moreover, the explicit solution is also expressed by the symmetric operator matrix, controllability matrix, and observability matrix.  The proposed approach does not require the coefficient matrices to be in arbitrary canonical form.  At the end of this paper, the numerical example is shown to illustrate the effectiveness of the proposed method.

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A New Solution to the Matrix Equation<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mrow><mml:mi>X</mml:mi><mml:mo>−</mml:mo><mml:mi>A</mml:mi><mml:mover accent="true"><mml:mi>X</mml:mi><mml:mo>¯</mml:mo></mml:mover><mml:mi>B</mml:mi><mml:mo>=</mml:mo><mml:mi>C</mml:mi></mml:mrow></mml:math>

A New Solution to the Matrix EquationX−AX¯B=C