kernel)*, if there exists $\mu>0$ such that all eigenvalues of ${\mathbf{C}}$ satisfy $$\Re(\lambda_j)\ge\mu\,, \qquad j=1,...,n\,.$$ While this notion was originally coined for operators in PDEs, such matrices are typically also called *positively stable*.

Comparing the spectrum of ${\mathbf{C}}$ and ${\mathbf{C}}_H$, it is well known that the maximum constants $\kappa$ and $\mu$ satisfy $\kappa\le\mu$. If all eigenvalues of ${\mathbf{C}}$ with $\Re(\lambda_j)=\mu$ are *non-defective*, then $f(t)$ decays at least with rate $\mu$. However, if ${\mathbf{C}}$ has a defective eigenvalue with $\Re(\lambda)=\mu$, then $f(t)$ decays “slightly slower”, i.e. with rate $\mu-{\varepsilon}$, for any ${\varepsilon}>0$ (see [@ArEr14 Proposition 4.5] and [@AAS15 Proposition 2.2] for details – applied to hypocoercive Fokker-Planck equations). Very recently this decay result has been improved as follows: In this case there is still a positive definite matrix ${\mathbf{P}}$, but it cannot be given by the simple formula , and  becomes $$\label{defective-decay}
 \|f(t)\|^2_{\mathbf{P}}\leq C (1+t^{2m}) e^{-2\mu t}  \|f^I\|^2_{\mathbf{P}}$$ for some $C>0$, where $m$ is the maximal defect of the eigenvalues of ${\mathbf{C}}$ with $\Re(\lambda_j)=\mu$. See [@ArEiWo] for more information.

Index of hypocoercivity {#sec-hypo-index}
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For the BGK models analyzed below we intend to construct convenient Lyapunov functionals of the form $\langle  f,{\mathbf{P}}f\rangle$, where the matrix ${\mathbf{P}}$ does not necessarily have to 