Lower bound on $x^T A x$ when A is non-symmetric with positive eigenvalues

Question

Let $x \in \mathbb{R}$ be a $m \times 1$ vector and $A \in \mathbb{R}$ be a non-symmetric $m \times m$ matrix with real and positive eigenvalues.

Is there a lower bound on $x^T A x$ of the form $\kappa ||x||^2$ with $\kappa >0$ ?

Answer 1

The answer is no. Consider the example

$$\begin{pmatrix}1&1\end{pmatrix}\begin{pmatrix}1&0\\-4&1\end{pmatrix}\begin{pmatrix}1\\1\end{pmatrix}=-2<0.$$