Hyperbolic fixed point of ODE

Question

Suppose we have a linear autonomous two dimensional ODE: \begin{equation} \frac{dx}{dt} = Ax \end{equation} for some matrix $A \in \mathbb{R}^{2 \times 2}$. Now we say a system is hyperbolic if the real part of the eigenvalues of $A$ are non-zero. Now wikipedia says the following: "Several properties hold about a neighborhood of a hyperbolic point, notably[2] A stable manifold and an unstable manifold exist" [https://en.wikipedia.org/wiki/Hyperbolic_equilibrium_point ] . But if I take $A=Id$ then all solutions diverge (except $0$ of course). So where is my stable subspace? Or analogue if I take a matrix with only negativ eigenvalues all solutions converge to $0$. Where is my unstable subspace? I dont understand this definition of hyperbolicity. Shouldn't it intuitively mean that one eignvalue is greater $0$ and one smaller $0$? Sorry Im really confused here.