reveal the sharp spectral gap of ${\mathbf{C}}$ (in the sense of Lemma \[lemma:Pdefinition\]). To this end we first give a definition of the structural complexity of a hypocoercive equation of the form $$\label{matrix-ODE}
  {\frac{{\,\operatorname{d}\!}}{{\,\operatorname{d}\!t}}}f +i\; {\mathbf{C}}_1 f = -{\mathbf{C}}_2 f\ ,\quad t\ge0\ .$$ Here we decomposed the matrix ${\mathbf{C}}\in{\mathbb{C}}^{n\times n}$ as ${\mathbf{C}}=i{\mathbf{C}}_1+{\mathbf{C}}_2$ with Hermitian matrices ${\mathbf{C}}_1$ and ${\mathbf{C}}_2$ with ${\mathbf{C}}_2\ge0$. In the special case ${\mathbf{C}}_1=0$, $(\ker {\mathbf{C}}_2)^\perp$ corresponds to the subspace of decaying solutions $f(t)$, and $\ker {\mathbf{C}}_2$ to the non-decaying subspace. In hypocoercive equations, the semigroup generated by the skew-Hermitian matrix $i{\mathbf{C}}_1$ may turn non-decaying directions into decaying directions, hence allowing for an exponential decay of all solutions. More precisely, we assume $$\label{hypocoercive:2}
 \exists \tau\in {\mathbb{N}}_0 \quad \text{and}\quad \exists \kappa>0 \ : \qquad \sum_{j=0}^\tau {\mathbf{C}}_1^j {\mathbf{C}}_2 ({\mathbf{C}}_1)^j \geq \kappa {\mathbf{I}}\ .$$

\[hyp-index\] For Hermitian matrices ${\mathbf{C}}_1$ and ${\mathbf{C}}_2$ with ${\mathbf{C}}_2\ge0$, the *hypocoercivity index* of the matrix ${\mathbf{C}}$ (and of the ODE ) is the smallest $\tau\in{\mathbb{N}}_0$, such that holds.

Clearly, $\tau=0$ corresponds to coercive matrices ${\mathbf{C}}$; i.e., those for which all eigenvalues of its Hermitian part $\tfrac12({\mathbf{C}}+{\mathbf{C}}^*)$ are strictly positive. A simple computation shows that this definition is invariant under a change of basis. We note that condition is 