Let $\pmb{\delta}=\pmb{\delta}^\triangle$ be an equilibrium point for the following ODE , \begin{align*} \frac{\partial \pmb{\delta}(t)}{\partial t}=\pmb{F}(\pmb{\delta}) \ with \ \ \pmb{F}(\pmb{\delta}^\triangle)=\pmb{0}, \end{align*} where $\pmb{F}: \mathbb{D}\to \mathbb{R}^d$ is continuously differentiable and $\mathbb{D}$ is a neighborhood of $\pmb{\delta}^\triangle$. Let \begin{align*} A=\frac{\partial \pmb{F}(\pmb{\delta})}{\partial \pmb{\delta}}\bigg|_{\pmb{\delta}=\pmb{\delta}^\triangle}, \end{align*} and $\chi$ denotes eigenvalue of $A$. Then, if $\chi_i$ are real and $\chi_i<0, i=1, 2, \cdots, d $ for all eigenvalues of $A$, $\pmb{\delta}^\triangle$ is an exponentially stable equilibrium point of this ODE.
Based on the above, whether we have the following inequality? \begin{align*} e^{\min\chi(A)t}\|\pmb{\delta}(0)-\pmb{\delta}^\triangle\| \leq \|\pmb{\delta}(t)-\pmb{\delta}^\triangle\| \leq e^{\max\chi(A)t}\|\pmb{\delta}(0)-\pmb{\delta}^\triangle\|. \end{align*}