I've got a linear system of equations in complex numbers:
$\mathbf{A}\mathbf{x}=\mathbf{b}$,
where $\mathbf{A}$ is fully-populated, $\mathbf{b}$ is not equal to zero, vector $\mathbf{x}$ is unknown and complex. Is it possible to decouple the real and imaginary parts of matrix $\mathbf{A}$ and treat them separately? On other words, is it possible to rewrite this equation as two equations with $Re(\mathbf{A})$ and $Im(\mathbf{A})$ separated? If yes, how can I do that?
The reason why I'm asking is that some elements of $\mathbf{A}$ have very different real and imaginary parts: $Im({A}_{ij})\approx10^{-9}Re({A}_{ij})$,
which I believe causes problems for numerical solvers.
Best Answer
Like Andrew said, this works by linearity. As an example, take \begin{equation} Ax=b \quad = \begin{bmatrix}1 & 0\\ 0 & i\end{bmatrix}\begin{bmatrix}1+i\\1+i\end{bmatrix}=\begin{bmatrix}1+i\\i-1\end{bmatrix} \end{equation} Then you can decouple the vector, i.e. $x=Re(x)+Im(x)$: \begin{equation} Ax=b \quad = \begin{bmatrix}1 & 0\\ 0 & i\end{bmatrix}\begin{bmatrix}1\\1\end{bmatrix}+\begin{bmatrix}1 & 0\\ 0 & i\end{bmatrix}\begin{bmatrix}i\\i\end{bmatrix}=\begin{bmatrix}1+i\\i-1\end{bmatrix} \end{equation} but you could also decouple the matrix, i.e. $A=Re(A)+Im(A)$: \begin{equation} Ax=b \quad = \begin{bmatrix}1 & 0\\ 0 & 0\end{bmatrix}\begin{bmatrix}1+i\\1+i\end{bmatrix}+\begin{bmatrix}0 & 0\\ 0 & i\end{bmatrix}\begin{bmatrix}1+i\\1+i\end{bmatrix}=\begin{bmatrix}1+i\\i-1\end{bmatrix} \end{equation}
edit: I thought this example would clarify how things work, but to again adopt what Andrew said: The Real part of the example becomes: \begin{align} &Re(A)Re(x)+Im(A)Im(x)=Re(b)\\ &\begin{bmatrix}1 & 0\\ 0 & 0\end{bmatrix}\begin{bmatrix}1\\1\end{bmatrix}+\begin{bmatrix}0 & 0\\ 0 & i\end{bmatrix}\begin{bmatrix}i\\i\end{bmatrix}=\begin{bmatrix}1\\-1\end{bmatrix} \end{align} Whereas the Imaginary part becomes: \begin{align} &Im(A)Re(x)+Re(A)Im(x)=Im(b)\\ &\begin{bmatrix}0 & 0\\ 0 & i\end{bmatrix}\begin{bmatrix}1\\1\end{bmatrix}+\begin{bmatrix}1 & 0\\ 0 & 0\end{bmatrix}\begin{bmatrix}i\\i\end{bmatrix}=\begin{bmatrix}i\\i\end{bmatrix} \end{align}