How can I show $\langle J^\epsilon x,x\rangle\leq-\alpha\|x\|^2\forall x\in\Bbb{R}^n$, if $\mathrm{Re}(\lambda)<0$, $J^\epsilon$ being a Jordan matrix

Question

$\forall \epsilon > 0$, there exists a basis $\mathcal{B}_\epsilon$ such that $J$, a Jordan form of a matrix $A$, is a block matriz $J^\epsilon = \operatorname{diag}\left[ J^\epsilon_1,...,J^\epsilon_k \right]$. $$ J^\epsilon = (P^\epsilon)^{-1}JP^\epsilon $$ of the form $\begin{bmatrix} \lambda_i & \epsilon&\cdots&\cdots&0\\ 0 & \lambda_i & \epsilon&\cdots&\vdots\\ \vdots &\ddots&\ddots&\ddots&\vdots\\ 0&\cdots&0 &\lambda_i& \epsilon\\ 0 &0&\dots&0&\lambda_i\end{bmatrix}$... $(\textbf{I didn't know how to make a matrix, sorry!!, I hope you could understand it})$ So, This is the problem: Let be $x' = J^\epsilon x$. If $Re(\lambda) < 0\; \forall \lambda\in J^\epsilon$, then, $\exists\; \alpha > 0$ such that $$\langle J^\epsilon x,x\rangle\leq -\alpha\|x\|^2 < 0\; \forall x \in \mathbb{R}^n$$ where $\phi^\epsilon_t = e^{J^\epsilon t}$ is the flow of the matrix $J^\epsilon$. SOME ADVICES, PLEASE!!