Assume you have an arbitrary, invertible $N\times N$ matrix $A$ and an $N$-dimensional vector $b$ I am looking for a Hermitian $2N\times 2N$ matrix $A'$ and a $2N$-dimensional vector $b'$ (obviously based on $A$ and $b$ respectively) such that a solution $x$ of the linear system $Ax=b$ can be read off from a solution $x'$ to the system $A'x=b'$.
I am not sure how to proceed, I thought of a way to make $A$ into a block of $A'$ but actually now I think that's bound to fail, as $A$ is arbitrary.
Best Answer
You can consider the system $$ \begin{bmatrix} 0&A\\ A^*&0\end{bmatrix} \begin{bmatrix} y\\ x\end{bmatrix} =\begin{bmatrix} b\\0\end{bmatrix}. $$ The "first coordinate" of this system is $Ax=b$. In other words, the coordinates $N+1,\ldots,2N$ of the solution are the solution of the original system.