depends only on the Hermitian part $\tfrac12 ({\mathbf{C}}^* + {\mathbf{C}})$ of matrix ${\mathbf{C}}$, such that for a Hermitian matrix ${\mathbf{C}}$ there is no loss of information.

But for non-Hermitian matrices it is more natural to use a modified norm: $$\|f\|^2_{\mathbf{P}}:= \langle f,{\mathbf{P}}f\rangle\,,$$ for some positive definite Hermitian matrix ${\mathbf{P}}\in{\mathbb{C}}^{n\times n}$, to be derived from ${\mathbf{C}}$. The derivative of $\|f\|^2_{\mathbf{P}}$ along solutions $f(t)$ of satisfies $${\frac{{\,\operatorname{d}\!}}{{\,\operatorname{d}\!t}}}\|f(t)\|^2_{\mathbf{P}}= -{\langle {f(t)}\ ,\, {({\mathbf{C}}^* {\mathbf{P}}+{\mathbf{P}}{\mathbf{C}})f(t)} \rangle} \,. $$ Then, $f^0\equiv 0$ is asymptotically stable, if there exists a positive definite Hermitian matrix ${\mathbf{P}}$ such that ${\mathbf{C}}^* {\mathbf{P}}+{\mathbf{P}}{\mathbf{C}}$ is positive definite. To determine the decay rate to $f^0$, and to choose ${\mathbf{P}}$ conveniently we shall use the following algebraic result.

\[lemma:Pdefinition\] For any fixed matrix ${\mathbf{C}}\in{\mathbb{C}}^{n\times n}$, let $\mu:=\min\{\Re(\lambda)|\lambda$ is an eigenvalue of ${\mathbf{C}}\}$. Let $\{\lambda_{j}|1\leq j\leq j_0\}$ be all the eigenvalues of ${\mathbf{C}}$ with $\Re(\lambda_j)=\mu$, only counting their geometric multiplicity. If all $\lambda_j$ ($j=1,\dots,j_0$) are non-defective, then there exists a positive definite Hermitian matrix ${\mathbf{P}}\in{\mathbb{C}}^{n\times n}$ with $$\begin{aligned}
 \label{matrixestimate1}
   {\mathbf{C}}^*{\mathbf{P}}+{\mathbf{P}}{\mathbf{C}}&\geq 2\mu {\mathbf{P}}\,.
  \end{aligned}$$ But ${\mathbf{P}}$ is not uniquely determined. Moreover, if all eigenvalues of ${\mathbf{C}}$ are non-defective, such matrices ${\mathbf{P}}$ satisfying  are given by $$\begin{aligned}
 \label{simpleP1} 
  {\mathbf{P}}:= \sum\limits_{j=1}^n b_j \,\overline{w_j}\otimes w_j^\top \,,
 \end{aligned}$$ where $w_j$ 