Change of real eigenvalues under symmetric perturbation

Question

Given a real matrix $A$ and a real diagonal matrix $D$.

Assuming $A$ has $k$ real eigenvalues. Will $A + D$ have $k$ real eigenvalues as well?

What if $D$ is symmetric?

Answer 1

The answer to both questions is no.

For instance, consider $$ A = \pmatrix{-2&-1\\1&1}, \quad D = \pmatrix{3&0\\0&0}. $$ $A$ has $2$ real eigenvalues, $A + D$ has none.