I have been given some lecture notes that explain how to solve a system of ODEs linearised around steady state. All the steps are clear except the following statements:
- $J(x^\ast)$ is real and symmetric <-- How can I show this?
- We can write all vectors in $\mathbb{R}^n$ as linear combinations of the eigenvectors of $J(x^\ast)$ <-- What is a good reference to understand this step?
- $x(t) = x^\ast + \sum_{n=1}^\infty \exp(\lambda_n t) c_n e_n$ <-- How can $\exp(\lambda_n t)$ be brought inside the summation? If it's a typo, how should it be corrected?
Can someone help clarify them please?
