Does eigenvalues bounded below imply matrix norms bounded below?

Question

If we have a sequence of matrices $A_n$ such that all of the eigenvalues are positive and bounded away from $0$, is it true that $|A_nx| \geq \lambda|x|$ for some $\lambda>0$?

Thank you

Answer 1

The matrices $A_n = \pmatrix{\dfrac{1}{n^2} & \dfrac{(n^2-1)^{3/2}}{n^2}\cr \dfrac{-\sqrt{n^2-1}}{n^2} & \dfrac{2n^2-1}{n^2}\cr}$ have all eigenvalues $1$, but satisfy $\|A_n x \| = \dfrac{1}{n}$ with $x = \pmatrix{1\cr 0\cr}$.