What more can be said about the eigenvalues (especially the spectrum) of the $N \times N$ matrix ${\bf M} = {\bf A} + {\bf A}^T$ in terms of $\bf A$ if $\bf A$ is not symmetric and its eigenvalues are not necessarily positive ($\bf A$ is not necessarily positive semi-definite)?

1. The eigenvalues of $\bf M$ are real since $\bf M$ is symmetric.
2. $\sum_{i=1}^{N} \lambda_i({\bf M}) = {\rm trace}({\bf M}) = 2~{\rm trace}({\bf A}) = 2\sum_{i=1}^{N} \lambda_i({\bf A})$