# Intuitive approach to the Jordan Normal form

I want to understand the meaning behind the Jordan Normal form, as I think this is crucial for 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 got the idea right? I mean, I have troubles to interpret 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 can explain to me, why we have to start a new block for each new linearly independent eigenvector that we can find? Why do we not have one block for each generalized eigenspace?

And what is the intuition behind the fact that the Jordan blocks that contain exactly or more than k+1 entries to the eigenvalue $\lambda$ are determined by $$\dim(\ker(A-\lambda I)^{k+1}) - \dim(\ker(A-\lambda I)^k)?$$

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