t}\ {\mathbf{Q}}\ e^{-{\mathbf{C}}t} {\,\operatorname{d}\!t}\ .$$ If ${\mathbf{Q}}$ is a positive definite symmetric matrix, then the unique solution ${\mathbf{P}}$ is also symmetric and positive definite. Moreover, the ${\mathbf{P}}$-norm of any solution $f(t)$ of satisfies $${\frac{{\,\operatorname{d}\!}}{{\,\operatorname{d}\!t}}}\|f(t)\|^2_{\mathbf{P}}= -{\langle {f(t)}\ ,\, {({\mathbf{C}}^* {\mathbf{P}}+{\mathbf{P}}{\mathbf{C}})f(t)} \rangle} 
    = -{\langle {f(t)}\ ,\, {{\mathbf{Q}}f(t)} \rangle}   
    \leq -\min \lambda^{\mathbf{Q}}_j \|f(t)\|^2
    \leq -\frac{\min \lambda^{\mathbf{Q}}_j}{\max \lambda^{\mathbf{P}}_j} \|f(t)\|^2_{\mathbf{P}}\ ,$$ where $\lambda^{\mathbf{Q}}_j$ and $\lambda^{\mathbf{P}}_j$ are the positive eigenvalues of the positive definite symmetric matrices ${\mathbf{Q}}$ and ${\mathbf{P}}$. This implies with $2\mu ={\min \lambda^{\mathbf{Q}}_j}/{\max \lambda^{\mathbf{P}}_j}$. However, only a suitable choice for ${\mathbf{Q}}$ would allow to recover the optimal decay rate as achieved in Lemma \[lemma:Pdefinition\].

The preceding discussion allows to characterize coercive and hypocoercive systems of linear ODEs (as well as matrices) according to the definition in the introduction: Equation  with matrix ${\mathbf{C}}$ is *coercive*, if the Hermitian part of ${\mathbf{C}}$ is positive definite, i.e. $$  \exists\kappa>0 \quad \text{such that} \quad {\mathbf{C}}_H:=\tfrac12 ({\mathbf{C}}+{\mathbf{C}}^*) \ge \kappa {\mathbf{I}}\,.$$ In this case, the trivial energy method (i.e. multiplying by $\overline{f}(t)^\top$ and using $\|f\|^2$ as a Lyapunov functional) shows decay of $f(t)$ with rate $\kappa$ and $c=1$. But this exponential rate is not necessarily sharp, e.g. for some non-Hermitian matrices ${\mathbf{C}}$.

Equation  with matrix ${\mathbf{C}}$ is *hypocoercive (with trivial 