5.  \[cond:compMatrix\] There exists a skew-Hermitian matrix ${\mathbf{K}}$ such that ${\mathbf{C}}_2 +[{\mathbf{K}},{\mathbf{C}}_1] ={\mathbf{C}}_2 +({\mathbf{K}}{\mathbf{C}}_1 -{\mathbf{C}}_1{\mathbf{K}})$ is positive definite.

Moreover, the smallest possible $\tau$ in \[cond:fullrank\] and \[cond:hypocoercive\] coincides; it is the hypocoercivity index of ${\mathbf{C}}$.

The equivalence of \[cond:fullrank\] and \[cond:hypocoercive\] (with the same $\tau$) follows from [@AAS15 Lemma 2.3]. The equivalence of \[cond:hypocoercive\]–\[cond:eigenvector\] follows from [@ArEr14 Lemma 2.3]. The equivalence of \[cond:eigenvector\] and \[cond:compMatrix\] follows by the same arguments as for real symmetric matrices in [@ShKa85 Theorem 2.5].

\[rem:hypo:cond:B1’\]

1.  In order to use condition \[cond:fullrank\] later on also for “infinite matrices” we give here an equivalent version:

    1.  \[cond:trivkernel\] There exists $\tau\in{\mathbb{N}}_0$ such that $\bigcap_{j=0}^\tau \ker\big(\sqrt{{\mathbf{C}}_2}{\mathbf{C}}_1^j\big) =\{0\}$.

2.  If $\tau\in{\mathbb{N}}_0$ is such that $$\label{rank-tau1} 
        \operatorname{rank}\{\sqrt{{\mathbf{C}}_2},{\mathbf{C}}_1 \sqrt{{\mathbf{C}}_2}, \ldots, {\mathbf{C}}_1^\tau \sqrt{{\mathbf{C}}_2}\} = \operatorname{rank}\{\sqrt{{\mathbf{C}}_2},{\mathbf{C}}_1 \sqrt{{\mathbf{C}}_2}, \ldots, {\mathbf{C}}_1^\tau \sqrt{{\mathbf{C}}_2}, {\mathbf{C}}_1^{\tau+1} \sqrt{{\mathbf{C}}_2}\} \ ,$$ then for all $k\in{\mathbb{N}}$ $$\operatorname{rank}\{\sqrt{{\mathbf{C}}_2},{\mathbf{C}}_1 \sqrt{{\mathbf{C}}_2}, \ldots, {\mathbf{C}}_1^\tau \sqrt{{\mathbf{C}}_2}\} = \operatorname{rank}\{\sqrt{{\mathbf{C}}_2},{\mathbf{C}}_1 \sqrt{{\mathbf{C}}_2}, \ldots, {\mathbf{C}}_1^{\tau+k} \sqrt{{\mathbf{C}}_2}\} \,.$$ Condition implies that the columns of ${\mathbf{C}}_1^{\tau+1} \sqrt{{\mathbf{C}}_2}$ are linear combinations of the columns of ${\mathbf{C}}_1^j \sqrt{{\mathbf{C}}_2}$, $j\in\{0,\ldots,\tau\}$. This implies that ${\mathbf{C}}_1^{\tau+k} \sqrt{{\mathbf{C}}_2}$ are linear combinations of the columns of ${\mathbf{C}}_1^j \sqrt{{\mathbf{C}}_2}$, $j\in\{k-1,\ldots,\tau+k-1\}$. Hence, for a hypocoercive matrix we have to gain with each added term in at least one rank until we 