In this post, the matrix exponential $$e^A = \sum_{k=0}^\infty \frac{1}{k!} A^k$$ of any symmetric matrix $A$ was proven to be positive-definite based on the quadratic function $$x^\top e^A x$$ where $x$ are weights that could sum to 1, I don't know.
In what sort of empirical applications is this weighted matrix exponential, $ x^\top e^A x$, even used?