I am studying a quantum algorithm for solving systems of linear differential equations (LDEs), which can be written in the following form: $$ \frac{d}{dt}(\vec{x})=M\vec{x}+\vec{b} $$ More specifically, I am interested in the case where there existis a matrix $A=M/c$ with $c$ an complex constant, such that $A$ is unitary. Recently I discovered that, when $A$ is also hermitian, I can alter the circuit in a really desirable way (less qubits, less operations and more precision).
Is there a way to create a hermitian matrix from a unitary one, without changing in a meaningful way the LDE system it is related to (the new system may contain more equations, but the original ones should be unaltered)? I know one can construct a hermitian matrix $U$ in this way: $$ U = \begin{bmatrix}0 & A \\ A^\dagger & 0\end{bmatrix} $$ A symple example, however, show that this construction is of no help, since it is associated with a completelly different system. Let's look at the simple harmonic oscillator: $$ \begin{bmatrix} \dot{s} \\ \dot{v}\end{bmatrix} = \begin{bmatrix}0 & 1 \\ -1 & 0\end{bmatrix} \begin{bmatrix} s \\ v\end{bmatrix} $$ I'll name the dummy variables $x_1$ and $x_2$. The new system, with the matrix $U$, will then be: $$ \begin{bmatrix} \dot{s} \\ \dot{v} \\ \dot{x_1} \\ \dot{x_2} \end{bmatrix} = \begin{bmatrix}0 & 0 & 0 & 1 \\ 0 & 0 & -1 & 0 \\ 0 & -1 & 0 & 0 \\ 1 & 0 & 0 & 0\end{bmatrix} \begin{bmatrix} s \\ v \\ x_1 \\ x_2\end{bmatrix} $$ This system is a completely new one, since the derivative of the old variables depend on the new, dummy ones. Also, setting $x_1=s$ and $x_2=v$, doesn't work, as it leads to false statetments.
I am inclined to think that this is impossible, but that is something I wasn't able to prove. Hence my question, is there a way to construct a hermitian matrix from a unitary one without affecting in a meaningful way the system it's related to?