identical to the matrix condition in Lemma 2.3 of [@ArEr14], which characterizes the hypoellipticity of degenerate Fokker-Planck operators of the form ${\mathbf{L}}f=\operatorname{div}({\mathbf{D}}\nabla f+{\mathbf{C}}xf)$ (using the matrix correspondence ${\mathbf{D}}={\mathbf{C}}_2$, ${\mathbf{C}}={\mathbf{C}}_1$). Hence, condition for the ODE and its hypocoercivity index can be seen as an analogue of the *finite rank Hörmander condition* for hypoelliptic and degenerate diffusion equations [@Ho67 Th. 1.1]. While the hypocoercivity index of degenerate parabolic equations determines the algebraic regularization rate (e.g. from $L^2$ into $H^1$, see Theorem A.12 in [@ViH06] and Theorem 4.8 in [@ArEr14]), its role in hypocoercive ODEs is not yet clear.

### Equivalent hypocoercivity conditions

Next, we collect several statements which are equivalent to condition . They will be useful for the analysis in §\[sec:Pmatrix\].

\[prop:equivalence\] Suppose that ${\mathbf{C}}_1\in{\mathbb{C}}^{n\times n}$ and ${\mathbf{C}}_2\in{\mathbb{C}}^{n\times n}$ are Hermitian matrices. Suppose furthermore that ${\mathbf{C}}_2$ is positive semi-definite. Then the following conditions are equivalent:

1.  \[cond:fullrank\] There exists $\tau\in{\mathbb{N}}_0$ such that $$\operatorname{rank}\{\sqrt{{\mathbf{C}}_2},{\mathbf{C}}_1 \sqrt{{\mathbf{C}}_2}, \ldots, {\mathbf{C}}_1^\tau \sqrt{{\mathbf{C}}_2}\} =n \ ,$$ which is often called *Kalman rank condition*.

2.  \[cond:hypocoercive\] The matrices ${\mathbf{C}}_1$ and ${\mathbf{C}}_2$ satisfy condition .

3.  \[cond:invariance\] No non-trivial subspace of $\ker {\mathbf{C}}_2$ is invariant under ${\mathbf{C}}_1$.

4.  \[cond:eigenvector\] No eigenvector of ${\mathbf{C}}_1$ lies in the kernel of ${\mathbf{C}}_2$.

