For two PSD matrices $A, B$ and a positive number $\alpha$, let's define $A \approx_{\alpha} B$ as $A \preceq \alpha B$ and $B \preceq \alpha A$, where $A \preceq B$ means $B - A$ is PSD. And if there exists a very small $\gamma$, such that $A \approx_{1 + \gamma} B$, we say $A \approx B$.
A professor told me that if $A \approx_{1 + \epsilon} B$ for a small enough $\epsilon$, then we have $A^2 \approx_{1 + O(\epsilon \kappa)^2} B^2$, where $\kappa$ is the condition number of $A$. If this bound is tight, then it means there exists a case such that $A \approx B$ but $A^2 \not \approx B^2$ when $\kappa(A)$ is very large.
These are my questions:
- Is the bound of the spectral approximation error of $A^2$ and $B^2$ tight? If it is tight, how to construct an example where $A \approx B$ but $A^2 \not \approx B^2$? If it is not, what would be a tighter bound, and how to derive it?
- Is there a sufficient/necessary condition to make $A \approx B$ but $A^2 \not \approx B^2$?