I have two non-symmetric matrices $A_1, A_2 \in \mathbb{R}^{n \times n}$ with negative real eigenvalues which are negative definite in the sense that their symmetric parts are negative definite, i.e. $$ x^T \frac{1}{2}(A_1 + A_1^T) x < 0 \quad \forall\, x \in \mathbb{R}^n \, . $$ I'm interested in the sum of the two matrices $A = A_1 + A_2$. It can be shown that $A$ is negative definite in the same sense and therefore all eigenvalues of $A$ have negative real parts: $$ \text{Re}(v^* A v) < 0 \quad \Rightarrow \quad \text{Re}(v^* A v) < v^*\, \text{Re}(\lambda)v \, . $$ Is it possible to make conclusions about the imaginary parts of the eigenvalues? Are there conditions for similar cases where you can make assertions about the imaginary parts? What additional information might be required to make progress? Or is it possible to construct a counter example, i.e. negative definie matrices $A_1$, $A_2$ with real eigenvalues such that $A = A_1 + A_2$ has non-zero imaginary parts?

The physical nature of the problem suggests that the eigenvalues have to be real because the matrices are constructed from a discretization of the heat equation. Unfortunately it has nonlinear (radiative) boundary conditions and the linearization terms destroy the symmetry of the system. Therefore I'm not sure whether the system is completely free of oscillations ($\rightarrow$ complex eigenvalues).


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