Generalized eigenspace decomposition of vector space

I was curious whether the direct sum of generalized eigenspaces in a finite dimensional vector space is the largest decomposition of this space invector spaces that are invariant under the endomorphism or are there counterexamples available?

• What would it mean that one decomposition is "larger" than another one? – Martin Argerami May 30 '13 at 22:56
• the question is whether all invariant subspaces are generalized eigenspaces? I see that we have generalized eigenspace => invariant subpace, but is the converse also true? – user66906 May 30 '13 at 22:57

Even in finite dimension, the number of invariant subspaces can be infinite. To see the most basic example, consider the identity endomorphism: it obviously has infinitely many invariant subspaces. Or consider $$V=\begin{bmatrix}1&0&0\\0&0&0\\0&0&0\end{bmatrix}$$ as an endomorphism of $\mathbb R^3$. Then, for each $t\in[0,1]$, the subspace $$X_t=\{(0,ct,c(1-t)):\ c\in\mathbb R\}$$ is invariant for $V$.