Center subspace ($E^c$)

For a finite-dimensional linearization $\dot{v}=Av$, the center subspace is the real invariant subspace associated with all generalized eigenvectors whose eigenvalues have zero real part. It includes zero eigenvalues and purely imaginary conjugate pairs. Generalized eigenspaces are needed when $A$ is not diagonalizable.

For $A=\mathrm{diag}(-1,0,2)$, it is the second coordinate axis. For a planar rotation matrix generator, it is the whole plane. The linear system alone gives no exponential contraction or expansion on this subspace; nonlinear terms can still cause growth or decay. A local center manifold is tangent to it at the equilibrium.