Consider two matrices $\mathbf{J} \in \mathbb{R}^{n \times n}$ and $\mathbf{L} = -\mathbf{L}^T \in \mathbb{R}^{n \times n}$. I am trying to upper bound the eigenvalues of their sum:
$$\mathbf{A} = \mathbf{J} + \mathbf{L}$$
in terms of the eigenvalues of $\mathbf{J}$. If it helps, the eigenvalues of $\mathbf{J}$ are distinct and have negative real part.
It is easy enough to show that the eigenvalues of the symmetric part of $\mathbf{A}$ are equal to the eigenvalues of the symmetric part of $\mathbf{J}$, but can something be said about the eigenvalues of $\mathbf{A}$ in terms of the eigenvalues of $\mathbf{J}$?
