I'm trying to shed some light on a recurrent problem I find while studying control systems. In many of the systems I work with, their stability depends on the eigenvalues of a matrix $B = U^{-1}A$, where $U_{ii} = u + \sum\limits_{j=1}^{n}|A_{ij}|, u \geq 0$, with $A,U,B \in \mathbb{R}^{n\times n}$. These can also be regarded as the solutions $\lambda$ of the GEP $Av = \lambda Uv$.
Particularly, the systems I study "work properly" (i.e., they are stable) when the eigenvalues of $B$ are positive. I know that this happens for sure if $A$ is symmetric and has positive eigenvalues since $U$ is positive-definite by definition. On the other hand, removing the symmetry condition on $A$ renders the positivity of its eigenvalues insufficient for the same conclusion.
What I wanted to understand, is if there is a "geometrical" approach that can give some insight into why symmetry is so important in this case, and tie this approach with the more formulaic one. I know that a matrix can be eigen-decomposed as $A = VDV^{-1}$, and that for symmetric matrices the eigenvectors are orthogonal, so $V^{-1} = V^T$ or in other words $V$ is a rotation. Is there a way to "visualize" the difference (in terms of the composition of $V^{-1}$ transformation - $D$ scaling - $V$ transformation - ...) between $A\cdot[e_1 e_2]$ and $U^{-1}A \cdot[e_1 e_2]$ in the following cases:
- $A$ symmetric with positive eigenvalues
- $A$ nonsymmetric with positive eigenvalues and $U^{-1}A$ with positive eigenvalues
- $A$ nonsymmetric with positive eigenvalues and $U^{-1}A$ with negative eigenvalues
- $A$ with positive and negative eigenvalues
that explains the "geometric significance" of the eigenvalues/eigenvectors of the composition $U^{-1}A$ in terms of the eigenvalues/eigenvectors of $U,A$, or on any other suitable set of vectors?