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Imagine that for a family of Hermitian $N\times N$ matrices $C(\alpha)$, $\alpha>0$, one knows that, for any $x\in\mathbb{C}^N\setminus\{0\}$, $\lim_{\alpha\to+\infty}\langle x,C(\alpha)x\rangle=+\infty$. (Here $\langle,\rangle$ is the standard scalar product in $\mathbb{C}^N$.)

Can one deduce that there is $\alpha_0>0$ such that $C(\alpha_0)$ is positive definite? Counterexamples?

In the case $N=1$ this is obviously true, but I cannot decide already when $N=2$.

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