# “Embedding” arbitrary square matrix into Hermitian

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.

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.