($j=1,\dots,n$) denote the left eigenvectors of ${\mathbf{C}}$, and $b_j\in{\mathbb{R}}^+$ ($j=1,\dots,n$) are arbitrary weights.

1.  This result was proven in [@ArEr14 Lemma 4.3] for real matrices ${\mathbf{C}}\in{\mathbb{R}}^{n\times n}$, and in [@AAC16 Lemma 2] for complex matrices ${\mathbf{C}}$. In particular, if ${\mathbf{C}}$ is a real matrix, then the inequality  of Lemma \[lemma:Pdefinition\] holds true for some real positive definite symmetric matrices ${\mathbf{P}}\in{\mathbb{R}}^{n\times n}$.

2.  For the extension of the above lemma to the case of defective eigenvalues see [@ArEr14 Lemma 4.3(i)] and [@AAS15 Prop. 2.2]. But the construction of ${\mathbf{P}}$ then involves also the generalized eigenvectors,

3.  The Lyapunov inequality is a special case of a linear matrix inequality. In a standard reference of system and control theory [@Boyd94], the problem of finding the maximal positive constant $\mu$ and a positive definite matrix ${\mathbf{P}}$ satisfying is formulated as a generalized eigenvalue problem, see [@Boyd94 §5.1.3]. The optimal value for the constant $\mu$ is pointed out, but the associated matrices ${\mathbf{P}}$ (like in our construction ) are not specified.

Now we consider examples, where all eigenvalues of ${\mathbf{C}}\in{\mathbb{C}}^{n\times n}$ are non-defective and have positive real parts. Then the origin is the unique and asymptotically stable steady state $f^0=0$ of $\eqref{ODE:general:n}$: Due to Lemma 