Given two square matrices $A$ and $B$. There are quite some results on the distance between the eigenvalues, e.g., 


$$
| \lambda\_A - \lambda\_B | \leq \| A - B \|\_F,
$$
where $A$ and $B$ are Hermitian (see [here for more](http://www.netlib.org/lapack/lawnspdf/lawn84.pdf)). I am looking for similar results for the argument of the eigenvalues, for example  

$$
| arg(\lambda\_A) - arg(\lambda\_B) | \leq \| A - B \|\_?
$$
One simple instance I have found: if invertible $A$ is normal, given the polar decomposition $A = UP$, then $| arg(\lambda\_A) - arg(\lambda\_U) | = 0$. Can this result be extended in terms of perturbation? 



Eigenvalue Argument Perturbation
Uhm, it does not hold even in $\mathbb{R}^{1\times 1}$. $A=\varepsilon, B = -\varepsilon$ gives $LHS=\pi$, $RHS = 2\varepsilon$.


