\[lemma:Pdefinition\], there exists a positive definite Hermitian matrix ${\mathbf{P}}\in{\mathbb{C}}^{n\times n}$ such that ${\mathbf{C}}^* {\mathbf{P}}+{\mathbf{P}}{\mathbf{C}}\geq 2\mu {\mathbf{P}}$ where $\mu = \min \Re(\lambda_j) >0$. Thus, the derivative of $\|f\|^2_{\mathbf{P}}= {\langle {f}\ ,\, {{\mathbf{P}}f} \rangle}$ along solutions of satisfies $$\begin{aligned}
  {\frac{{\,\operatorname{d}\!}}{{\,\operatorname{d}\!t}}}\|f(t)\|^2_{\mathbf{P}}&\leq -2\mu \|f(t)\|^2_{\mathbf{P}}\qquad \text{with} \quad \mu = \min \Re(\lambda_j),
 \end{aligned}$$ which implies $$\label{ODE-decay}
 \|f(t)\|^2_{\mathbf{P}}\leq e^{-2\mu t} \|f^I \|^2_{\mathbf{P}}\,,\qquad t\ge0\,.$$ Let $\lambda^{\mathbf{P}}_j$ $(j=1,\ldots,n)$ denote the positive eigenvalues of the positive definite Hermitian matrix ${\mathbf{P}}$ being ordered by magnitude such that $0 <\lambda^{\mathbf{P}}_1 \leq\ldots \leq\lambda^{\mathbf{P}}_n$. Then the matrix inequality $\lambda^{\mathbf{P}}_1 {\mathbf{I}}\leq {\mathbf{P}}\leq\lambda^{\mathbf{P}}_n {\mathbf{I}}$ implies the equivalence of norms $$\lambda^{\mathbf{P}}_1 \|v\|^2 \leq \|v\|_{\mathbf{P}}^2 \leq \lambda^{\mathbf{P}}_n \|v\|^2 \qquad \forall v\in{\mathbb{C}}^n \,.$$ Thus the decay in ${\mathbf{P}}$-norm  translates into a decay in the Euclidean norm $$\label{ODE:exponentialDecay}
 {\| {f(t)} \|}^2\leq c e^{-2\mu t} {\| {f^I} \|}^2\,,$$ with the constant $c=\lambda^{\mathbf{P}}_n /\lambda^{\mathbf{P}}_1 \ge1$, i.e. the condition number of ${\mathbf{P}}$. Note that $c=1$ if and only if ${\mathbf{P}}={\mathbf{I}}$.

In a popular textbook on linear systems theory [@Hes09], the exponential decay  is obtained as follows [@Hes09 §8.5]: For a stable matrix $-{\mathbf{C}}$ (i.e. all eigenvalues of $-{\mathbf{C}}$ have negative real part) and a matrix ${\mathbf{Q}}$, the unique solution ${\mathbf{P}}$ of Lyapunov’s equation $$  {\mathbf{C}}^* {\mathbf{P}}+{\mathbf{P}}{\mathbf{C}}= {\mathbf{Q}}$$ is given by $${\mathbf{P}}:= \int_0^\infty e^{-{\mathbf{C}}^* 