I have the system \begin{align} \frac{dx}{dt} &= y + x^2\\ \frac{dy}{dt} &= -y + x^2 \end{align} with an equilibrium point at the origin
I've found the Jacobian matrix, and that
eigenvalue $1 = 0$
eigenvector $1 = (1, 0)$
eigenvalue $2 = -1$
eigenvector $2 = (1, -1)$
How do I find the linearised system from these eigenvectors? And how would I do it for general eigenvectors other than these ones? Thank you