Suppose $P(x): \mathbb{R} \to \mathbb{R}^{n \times n}$ is always a positive definite matrix, does it imply that the infimum (over $\mathbb{R}$) of the minimum eigenvalue of $P(x)$ is always positive?, that is, $\inf_{x \in \mathbb{R}} \{\lambda_{{\rm min}}(P(x)) \} > 0$ In a mathematical way:
Is the following conclusion correct?
\begin{equation}
\forall x \in \mathbb{R},\;P(x) \succ 0 \implies \inf_{x \in \mathbb{R}} \{\lambda_{{\rm min}}(P(x)) \} > 0
\end{equation}
Please be careful with the infimum.
Please go to the other similar question here, which is exactly the real (and more difficult in my opinion) question that I want to ask.
Best Answer
No. Consider $n=1$ and $P(x)=e^x$ or in general, $P(x)=e^{xI_n}$.