Eigenvalues of $A$ and $A + A^T$

This question has popped up at me several times in my research in differential equations and other areas:

Let $A$ be a real $N \times N$ matrix. How are the eigenvalues of $A$ and $A + A^T$ related?
Of course, if $A$ is symmetric, the answer is easy: they are the same up to factor or $2$, since then $A + A^T = 2A$. But if $A \ne A^T$?

I’m particularly interested in the question of the real parts of the eigenvalues. How are the real parts of the eigenvalues of $A$ related to the (necessarily) real eigenvalues of $A + A^T$?

Answers for complex matrices appreciated as well.

Any references, citings, or explanations at any level of detail will be appreciated.

Thanks in advance.

Solutions Collecting From Web of "Eigenvalues of $A$ and $A + A^T$"

This question was answered here, see also comments to its “closed-as-duplicate” post, especially ones by Terry Tao.

Here is Tao’s comment from, now deleted, post 2:

“Note that if A is strictly upper triangular, then its eigenvalues are all zero, whereas $A+A^T$ is an arbitrary symmetric matrix with zero diagonal, which constrains the trace of the matrix but otherwise imposes almost no conditions on the spectrum whatsoever (the only other constraint I can see is that the matrix cannot be rank one). So, apart from the trace $tr(A+A^T)=2tr(A)$, there appears to be essentially no relationship.”

EDIT: Let $\lambda\in spectrum(A)$; then there is $\mu \in spectrum(A+A^T)$ s.t. $|\lambda-\mu|\leq ||A||_2$ (spectral norm).

Proof: Since $A+A^T$ is real symmetric, according to the Bauer–Fike Theorem, there is $\mu\in spectrum(A+A^T)$ s.t. $|\lambda-\mu|\leq ||-A^T||_2=||A||_2$. cf. http://en.wikipedia.org/wiki/Bauer%E2%80%93Fike_theorem

A special case is when $A$ is a unitary matrix: eigenvalues of $A$ lie on unit circle in complex plane. If $e^{i\theta}$ is an eigenvalue of $A$, then $2 \cos(\theta)$ is an eigenvalue of $A + A^T$. Similarly, $2i\sin(\theta)$ is an eigenvalue of $A – A^T$.