TITLE: An intuitive approach to the Jordan Normal form
QUESTION [53 upvotes]: I want to understand the meaning behind the Jordan Normal form, as I think this is crucial for a mathematician.
As far as I understand this, the idea is to get the closest representation of an arbitrary endomorphism towards the diagonal form. As diagonalization is only possible if there are sufficient eigenvectors, we try to get a representation of the endomorphism with respect to its generalized eigenspaces, as their sum always gives us the whole space. Therefore bringing an endomorphism to its Jordan normal form is always possible.
How often an eigenvalue appears on the diagonal in the JNF is determined by its algebraic multiplicity.  The number of blocks is determined by its geometric multiplicity. Here I am not sure whether I've got the idea right. I mean, I have trouble interpreting this statement.

What is the meaning behind a Jordan normal block and why is the number of these blocks equal to the number of linearly independent eigenvectors?

I do not want to see a rigorous proof, but maybe someone could answer for me the following sub-questions.

(a) Why do we have to start a new block for each new linearly independent eigenvector that we can find?
(b) Why do we not have one block for each generalized eigenspace?
(c) What is the intuition behind the fact that the Jordan blocks that contain  at least $k+1$ entries of the eigenvalue $\lambda$ are determined by the following? $$\dim(\ker(A-\lambda I)^{k+1}) - \dim(\ker(A-\lambda I)^k)$$

REPLY [4 votes]: In these notes I give a "middlebrow" approach to invariant subspaces and canonical forms.  Middlebrow means here that it is a bit more sophisticated than what you would encounter in a first linear algebra course -- in particular I work over an arbitrary field and then specialize to the algebraically closed case -- but that it stays in the setting of linear algebra rather than module theory: especially, the idea of abstract isomorphism (of modules) is never used but only similarity (of matrices).  Nevertheless this approach would generalize to give the structure theorem for finitely generated modules over a PID with little trouble.
My perspective is that of understanding invariant subspaces more generally and finding all of them, if possible (I pursue this problem a bit more doggedly than in most of the standard treatments I know).  The key result is the Cyclic Decomposition Theorem in Section 5, which says that given any endomorphism $T$ on a finite dimensional vector space $V$, one can write $V$ as a direct sum of subspaces stabilized by $T$ and on which the minimal polynomial of $T$ is primary, i.e., a power of an irreducible polynomial.  This is the appropriate generalization of "generalized eigenspace" to the non-algebraically closed case.  The Jordan canonical form follows easily and is discussed in Section 6.  In my terminology, JCF exists if and only if the minimal polynomial is split, i.e., is a product of linear factors over the ground field.  
Before I wrote these notes it had been many years since I had had to think about JCF, so for me at least they are meant to give a simple(st) conceptual explanation of JCF.
There are certainly other approaches.  Just to briefly point at one more: JCF is a nice application of the Chinese Remainder Theorem for modules: see e.g. Section 4.3 of my commutative algebra notes.  From this perspective the natural generalization would be the concept of primary decomposition of a module (which unfortunately I do not discuss in my commutative algebra notes, but most of the standard references do): what this is for becomes more clear when one studies algebraic geometry.