Talk:Jordan–Chevalley decomposition

Description of the unipotent part in terms of the Jordan normal form

The claim that the unipotent part is the Jordan normal form with the diagonal elements set to 1 is false.

Consider the matrix

${\begin{bmatrix}2&1\\0&2\end{bmatrix}}$

This is in Jordan normal form and its unipotent part is

${\begin{bmatrix}1&0.5\\0&1\end{bmatrix}}$

The current text suggests that its unipotent part is

${\begin{bmatrix}1&1\\0&1\end{bmatrix}}$

which is false.

Jeremy Henty (talk) 14:47, 17 April 2009 (UTC)

I have fixed the description

Jeremy Henty (talk) 14:56, 17 April 2009 (UTC)

Connection to Jordan normal form

JackSchmidt, you have deleted the paragraph Jordan–Chevalley decomposition#Connection to Jordan normal form in this edit with the edit summary "rm dupe of previous paragraph". I am afraid I do not understand the comment or why. Can you please explain your reasons? Do you believe the paragraph is incorrect? This is how Jordan normal form was taught to me a while ago. I would think the connection is a good way how to make the article and its importance understandable to mere mortals who may not be familiar with that particular corner of abstract algebra and want to learn from Wikipedia. Jmath666 (talk) 18:59, 11 January 2009 (UTC)

Sorry for the short edit summary. It meant "remove duplication of previous paragraph." Your paragraph was quite correct and important. I removed it because it just repeated the previous paragraph. It had a few style issues (no wikilinks in headings, maybe some reordering of phrases or something), so I kept the previous paragraph instead. The paragraph you added contains material that is both correct and important to this article, but that was already in the article. I removed one copy of the information (chosen based on which one was easier to remove).

If the current copy is too hard to see, we could maybe make it more prominent. I checked that at least the Jordan normal form link was there, so it should have been a big blue word. However mentioning it in the table of contents or the intro paragraph might be a good idea (I don't know, but I am not opposed).

As far as the teaching part goes, I definitely agree. I only learned the J-C decomposition when I realized it was the Jordan normal form, and then I understood the Jordan normal form quite a bit better. In particular, I think the additive and multiplicative versions of the J-C decomposition describe in a very nice way the two essential pieces of the Jordan block, and in some sense explain the notation. JackSchmidt (talk) 19:12, 11 January 2009 (UTC)

Indeed I did not notice the previous paragraph it was still too obscure. At a second look, to make the connection at least matrices should be uppercase, the change of basis should be explicitly mentioned, etc. Also the connection should be mentioned up front before all the big abstract algebra words that put off people who live in euclidean spaces. I recall my abstract algebra class did JNF over a general field not just reals, is then any difference between JNF and J-C decomposition at all? On another subject, there is a version in Banach spaces (not finite dimensional, nilpotent replaced by topological nilpotent, important for spectral theory). It seems the article in the present form takes a bit onesided classical abstract algebra view that may limit its usefulness. Thanks, Jmath666 (talk) 19:38, 11 January 2009 (UTC)

Even Jitse did not notice the two paragraphs say essentially the same thing... the connection should really be made more obvious for teaching reasons. Jmath666 (talk) 19:42, 11 January 2009 (UTC)

Sounds good. I'll try to merge your version in (that is, restore it, and then correct the MOS things, and mention it in the intro). There are differences, but they are mostly of emphasis. JNF is something you can change bases to get to, so is a factorization A=XJX^-1, but JCD is a factorization SU that is most obvious when expressed as a factorization of J. JackSchmidt (talk) 19:49, 11 January 2009 (UTC)

Ok, I added it back. To your paragraph I added the multiplicative version. I made section headings for the three usages. Does this look good?

I suspect it would be a good idea to write a fourth section on algebraic groups (factoring into subgroups), since then we could do: (1) matrices, (2) linear operators on fin. dim. and algebraic groups, (3) linear operators on banach spaces and banach algebras. I think Dunford's result is not just on individual operators, but on whole sets of them, which would fit in well with the JC decomposition of algebraic groups. JackSchmidt (talk) 20:04, 11 January 2009 (UTC)

Nice! The matrix example makes both cases so much clearer. I might take up the Banach space case at some point. Jmath666 (talk) 00:59, 12 January 2009 (UTC)

terrible writing

Counterexample to existence over an imperfect field - this and other parts are terribly writen. This is no way to produce a useful example, needs to be clarified much better. It has been added by some user who has since been banned for arrogance, and I cant say I am surprised, article needs to be more readable. Also, added proof in abstract terms is way too poorly explained. This is just terrible writing... 213.198.204.144 (talk) 18:09, 20 November 2023 (UTC)