Let $A$ be a matrix. If $A$ is "almost" equal to $A^\*$, it follows from an argument of continuity that the eigenvalues of $A$ are "almost" real. Same argument can be made for $A$ "almost" $-A^\*$, in which case the eigenvalues are "almost" purely imaginary.
Question: Is there a way of getting a quantitative estimate on the relative or absolute value of the imaginary and real parts of the eigenvalues?



Estimate on the real and imaginary parts of eigenvalues
Suppose $A=B+C$ where $B$ is self adjoint. Then you can diagonalize $B$, and then by applying Gersgorin's criterion, you get the following: if $\{\lambda\_i\}$ are the eigenvalues of $B$ and $\hat \lambda$ is an eigenvalue of $A$, then
there is an $i$ so that $|\hat \lambda-\lambda\_i|\leq \|C\|\_2$.
(Here, $\|C\|\_2$ is the Frobenius norm of $C$.)


