Let $A$ be an $n\times n$ real symmetric matrix, and let $E$ be a real matrix. Is it true that if the perturbation matrix $E$ is small in some norm, then the eigenvalues of $\hat A := A+E$ are all real? What if we have more properties on $A$ and $E$: $A$ is diagonalizable and $E$ is skew symmetric?
Note: I have seen two interesting lines of discussions on the eigenvalues of a symmetric matrix with perturbation here and here, but there is no discussion on the reality of eigenvalues. Thanks.