# Stable subspace, unstable subspace, centre subspace: Of which kind of stability are we talking?

Let $x'(t)=Ax(t)$ be a linear ODE system.

Then the span of the eigenvectors belonging to eigenvalues with negative real-part is called the stable subspace, the subspace spanned by the eigenvectors of eigenvalues of positive real-part is called unstable subspace and the subspace spanned by the eigenvectors belonging to eigenvalues with zero real-part is called centre subspace.

1. What kind of stability is the stable subspace of?

2. What kind of unstability (as negation of what kind of stability) is the unstable subspace of?

3. Is the centre subspace stable or unstable? Are there situation in which it is stable (and which kind of stability then)?

1. Globally asymptotically stable: solutions with initial values anywhere in this subspace converge to $0$.
All of this follows by considering the compression of $A$ to the relevant subspace, since the orbits beginning there stay in it. Reference: G. Teschl, ODE and Dynamical Systems.