I am afraid this is not a good setting to use the SVD. The point is that here the only natural transformation to make is changing basis for $x$, i.e., $x\to Ux$, and this leads to a transformation of the associated matrix of the form $M \mapsto UMU^{-1}$. So the basis-independent characteristics of the system will depend on the canonical form of $M$ under that kind of transformations, i.e., the Jordan form. Even if you want to restrict to orthogonal changes of basis, you'd get the canonical form under the action of $O(n)$ as $M\to UMU^T$ (which, incidentally, is quite messy for a non-symmetric $M$).

The SVD, instead, is the canonical form under the action of $O(n)\times O(n)$ as $M\mapsto UMV^T$, which would correspond in your problem to choosing different bases for $x$ and for $\dot{x}$, and I am afraid that it makes little sense to do it in this setting.

I am afraid this is not a good setting to use the SVD. The point is that here the only natural transformation to make is changing basis for $x$, i.e., $x\to Ux$, and this leads to a transformation of the associated matrix of the form $M \mapsto UMU^{-1}$. So the basis-independent characteristics of the system will depend on the canonical form of $M$ under that kind of transformations, i.e., the Jordan form. Even if you want to restrict to orthogonal changes of basis, you'd get the canonical form under the action of $O(n)$ as $M\to UMU^T$, which is the Schur form.

The SVD, instead, is the canonical form under the action of $O(n)\times O(n)$ as $M\mapsto UMV^T$, which would correspond in your problem to choosing different bases for $x$ and for $\dot{x}$, and I am afraid that it makes little sense to do it in this setting.