Consider the linear ODE \begin{align} \dot x = J x, \: x\in \mathbb{R}^n \end{align} with the Jacobian $J$. Page 35 of Ordinary Differential Equations with Applications by Chicone asserts that the space $\mathbb{R}^n$ can always be decomposed as a direct sum of linear subspaces: the stable eigenspace (stable manifold) corresponding to the eigenvalues of $J$ with negative real parts, the unstable eigenspace (unstable manifold) corresponding similarly to the eigenvalues of $J$ with positive real parts, and the center eigenspace (center manifold) corresponding to the eigenvalues with zero real parts.
My question is: what are the implications of $J$ being defective? For example, suppose \begin{align} J = \begin{pmatrix} -1 & 1 \\ 0 & -1 \end{pmatrix} \end{align} which has the eigenvalue of $-1$ with algebraic multiplicity of $2$, but the geometric multiplicity of $1$. The eigenspace is spanned by $[1,0]$. I was wondering what happens to the Hartman-Grobman Theorem in this case and how the space is partitioned.
Thank you, in advance, for your comments.
What happens is that the existence of a topological conjugacy between linear equations whose matrices have only eigenvalues with nonzero real part depends solely on the number of eigenvalues with positive (or negative) real part. More precisely:
The proof becomes somewhat simple only after introducing specific norms related to $A$ and $B$.