reach full rank, i.e. space dimension $n$. Thus, for hypocoercive matrices its hypocoercivity index is bounded from above by the dimension of $\ker{\mathbf{C}}_2$ (or equivalently corank of ${\mathbf{C}}_2$).

In [@ViH06 Remark 17] the connections of the above conditions to *Kawashima’s nondegeneracy condition* for the study of degenerate hyperbolic-parabolic systems [@Ka87] and *Hörmander’s rank condition* for hypoelliptic equations [@Ho67] are noted.

For real symmetric matrices ${\mathbf{C}}_1, {\mathbf{C}}_2\in {\mathbb{R}}^{n\times n}$ with ${\mathbf{C}}_2\geq 0$, condition \[cond:eigenvector\] is equivalent to the condition that ${\mathbf{C}}:=i{\mathbf{C}}_1 +{\mathbf{C}}_2$ has only eigenvalues with positive real part, see [@ShKa85 Theorem 1.1]. And the latter statement is equivalent to the exponential stability of . Using Proposition \[prop:equivalence\], we shall now prove a similar statement for Hermitian matrices:

\[lem:hypocoercive\] Hermitian matrices ${\mathbf{C}}_1$ and ${\mathbf{C}}_2$ with ${\mathbf{C}}_2\ge0$ satisfy condition  if and only if all eigenvalues $\lambda_{{\mathbf{C}}}$ of ${\mathbf{C}}:=i{\mathbf{C}}_1 +{\mathbf{C}}_2$ have positive real part $\Re (\lambda_{{\mathbf{C}}}) >0$.

To show Lemma \[lem:hypocoercive\] for Hermitian matrices, we will follow the proofs of [@UmKaSh84 Prop. 2.4] and [@ShKa85 Lemma 3.2] for real symmetric matrices.

First, we show that condition  implies that all eigenvalues $\lambda_{{\mathbf{C}}}$ of ${\mathbf{C}}:=i{\mathbf{C}}_1 +{\mathbf{C}}_2$ have positive real part $\Re (\lambda_{{\mathbf{C}}}) >0$: Let $\phi$ be an eigenvector of ${\mathbf{C}}$ corresponding to an eigenvalue $\lambda$, 