Solvability of the linear matrix equation $AX+B\overline{X}=C$? I search solvability properties of the matrix equation
$$A X + B \overline{X} = C$$ 
where $A, B, C, X \in \mathbb C^{n \times n}$. Can anybody help me?
 A: We have the linear matrix equation in $\mathrm X \in \mathbb C^{n \times n}$
$$\rm A X + B \overline{X} = C$$ 
where $\mathrm A, \mathrm B, \mathrm C \in \mathbb C^{n \times n}$ are given. Let $\mathrm M_{\text{re}}$ and $\mathrm M_{\text{im}}$ be the real and imaginary parts of $\mathrm M \in \mathbb C^{n \times n}$, respectively. Hence, the original linear matrix equation can be written as follows
$$\left( \left( \mathrm A_{\text{re}} + \mathrm B_{\text{re}} \right) \mathrm X_{\text{re}} + \left( -\mathrm A_{\text{im}} + \mathrm B_{\text{im}} \right) \mathrm X_{\text{im}} \right) + i \left( \left( \mathrm A_{\text{im}} + \mathrm B_{\text{im}} \right) \mathrm X_{\text{re}} + \left( \mathrm A_{\text{re}} - \mathrm B_{\text{re}} \right) \mathrm X_{\text{im}} \right) = \mathrm C_{\text{re}} + i \, \mathrm C_{\text{im}}$$
which yields two linear matrix equations in $\mathrm X_{\text{re}}, \mathrm X_{\text{im}} \in \mathbb R^{n \times n}$
$$\begin{array}{rl} \left( \mathrm A_{\text{re}} + \mathrm B_{\text{re}} \right) \mathrm X_{\text{re}} + \left( \mathrm B_{\text{im}} - \mathrm A_{\text{im}} \right) \mathrm X_{\text{im}} &= \, \mathrm C_{\text{re}}\\ \left( \mathrm A_{\text{im}} + \mathrm B_{\text{im}} \right) \mathrm X_{\text{re}} + \left( \mathrm A_{\text{re}} - \mathrm B_{\text{re}} \right) \mathrm X_{\text{im}} &= \, \mathrm C_{\text{im}}\end{array}$$
In matrix form,
$$\begin{bmatrix} \mathrm A_{\text{re}} + \mathrm B_{\text{re}} & \mathrm B_{\text{im}} - \mathrm A_{\text{im}}\\ \mathrm A_{\text{im}} + \mathrm B_{\text{im}} & \mathrm A_{\text{re}} - \mathrm B_{\text{re}}\end{bmatrix} \begin{bmatrix} \mathrm X_{\text{re}}\\ \mathrm X_{\text{im}}\end{bmatrix} = \begin{bmatrix} \mathrm C_{\text{re}}\\ \mathrm C_{\text{im}}\end{bmatrix}$$
Once $\mathrm X_{\text{re}}$ and $\mathrm X_{\text{im}}$ have been found, the solution to $\rm A X + B \overline{X} = C$ is simply $\mathrm X = \mathrm X_{\text{re}} + i \,\mathrm X_{\text{im}}$.
A: If you have a way of finding generalised inverses then this is not too difficult. 
Let $A'$ be such that $AA'A=A $. 
Then your equation is solvable iff $
\[AA'(C-B\overline {X})=C-B\overline {X}.\]$
Rearranging, this is solvable iff $
\[(I-AA')C=(I-AA')B\overline {X}.\] $
Now it is easy to find the solvability conditions of this last equation and its general solution.
