Jordan etc.:
What do these -say-?
Given an operator $A \in \gl V$:

algebraically: decompose
[Jordan-Chevalley]
$A$ decomposes uniquely as $A=A_{ss} + A_{nil}$,
where $A_{ss}$ is semi-simple, $A_{nil}$ is nilpotent,
and they're both expressable as a polynomial in $A$;
in particular, they commute.
[However, this polynomial depends on $A$ -- this is kinda mysterious
 to me, and feels like minimal/characteristic poly.]
Pf: $A$ generates a commutative subalgebra;
the nilpotent elements form an ideal (b/c it's commutative,
so if $B$ is nilpotent, so is $BC$: $(BC)^k = B^kC^k$)
and for some reason (here's the mystery), there's a natural Levi
compliment. In particular, you can project $A$ onto the ss and nil
parts, and these images are polynomials in $A$.
So this is not an insane statement.

top: flags
Schur decomposition:
$A$ stabilizes a flag
[NB: $A$ is regular iff it stablizes a -unique- flag;
 the generic element is regular semisimple, hence diagonalizable with
 distinct eigenvalues.]
many pfs:
- has an eigenvector (Axler)
- determinant has a root (standard)
- action on $\gl/\b$ has a fixed point (topological)

lin alg: gen. eigen, jordan
In a basis subordinate to a flag it stablizes,
$A_{ss}$ is diagonal and $A_{nil}$ is strictly upper triangular.

$V$ decomposes as -generalized- eigenspaces of $A$;
this is natural, and in any basis subordinate to the splitting, it's
block diagonal.

More detailed study of nilpotent operators on a space let us refine
that to get -Jordan blocks- for $A$ on a gen.eigenspace;
sexy as these blocks are, they're not very natural and are an artifact
of the way we write matrices; the kernel/image flags are what are
natural.

In summary:
Given an operator $A$:
- it splits uniquely into commuting ss and nil parts
  [indeed, the whole algebra it generates does]
- it stabilizes at least 1 flag (generically only 1,
  in which case it's called regular)
- $V$ splits into generalized eigenspaces for $A$
- in a basis subordinate to the flag it stabilizes
  and to the generalized eigenspaces,
  it's both upper triangular and block diagonal
- detailed study of nilpotent operators yields the Jordan form
