Solvability of the linear matrix equation $AX+B\overline{X}=C$?

Question

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?

Answer 1

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 $\mathrm X = \mathrm X_{\text{re}} + i \,\mathrm X_{\text{im}}$.

Answer 2

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.