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.


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*What kind of stability is the stable subspace of?

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

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

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*Globally asymptotically stable: solutions with initial values anywhere in this subspace converge to $0$.

*Negation of any kind of stability: the solutions with initial values in this subspace are unbounded, eventually leaving every compact set.

*Center is (globally) Lyapunov stable, not asymptotically stable.
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.
