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Can anyone suggest a reference that discusses the history of the Jordan canonical form? In particular, I am interested in:

  1. When and how was it first stated? (I understand it was independently stated by Jordan and Weierstrass.)

  2. When was it thought of in terms of partial fractions? (it seems this should have been the historically first version.)

  3. When was it thought of in terms of projections (i.e. orthogonal idempotents)?

  4. When was it thought of in terms of representations of the polynomial algebra $\mathbb{C}[x]$?

  5. When was the relationship with abelian groups recognized, and more generally modules over Euclidean domains?

  6. What is the history of the fundamental theorem of finitely generated modules over a PID?

  7. When was the Jordan-Chevalley decomposition first stated? (Unique representation of a matrix as a sum $S+N$ where $S$ is semisimple, $N$ is nilpotent, and $SN=NS$.) I assume this is due to Chevalley?

Apparently there is a PhD thesis about this by Frédéric Brechenmacher, but I don't have access to it.

Thanks.

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    $\begingroup$ The thesis you mention can be obtained freely from the TEL server: tel.archives-ouvertes.fr/tel-00142786 $\endgroup$
    – user9072
    Dec 18, 2013 at 15:42
  • $\begingroup$ @Drew: You've asked many questions here, so a short answer is impossible. But you might look at my response to an earlier MO question mathoverflow.net/questions/108402 and the reference to Couty et al. Also, searching MO for "Jordan-Chevalley" is useful. $\endgroup$ Dec 18, 2013 at 16:37
  • $\begingroup$ see also this post hsm.stackexchange.com/questions/5230/… I believe the context of the discovery was the invariant theory of pencils of quadratic forms. $\endgroup$ Jan 13, 2017 at 0:13

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As I pointed out in my comment, there are too many questions listed here. Maybe I can clarify the term "Jordan-Chevalley decomposition" in the last one. Besides the arXiv post by Danielle Couty and her colleagues which I gave a link to, it's worth looking at the revised published version and its substantial reference list. The article was published in the SMF journal Gazette des Mathematiciens, freely available online here. They look closely at the role Chevalley played, beyond the basic Jordan decomposition of matrices. But it should be emphasized that there are two related frameworks for this kind of decomposition:

1) In the study of Lie algebras over an arbitrary field of characteristic 0, indepndent from the original study of Lie groups over $\mathbb{R}$ or $\mathbb{C}$ by E. Cartan and others, semisimple Lie algebras are of special interest. From the nondegeneracy of the Killing form one sees that the Lie algebra is isomorphic to its derivation algebra. Moreover, in the Jordan decomposition of an adjoint operator, the unique semisimple and nilpotent parts belong to elements of the Lie algbra. Going further, one proves Weyl's complete reducibility theorem and then as a corollary the preservation of the intrinsic Jordan decompsition under any finite dimensional linear representation of the Lie algebra.

Chevalley influenced much of this development, in his books and seminar during the 1950s, but it's hard to document precisely. The Bourbaki group (of which he was an early member) began in 1960 to publish their treatise on Lie groups and Lie algebras. In Chapter I on Lie algebras, there is a crucial proof that complete reducibility in characteristic 0 implies the preservation of Jordan decomposition under homomorphisms: see $\S6$, no. 3, Thm. 3. These results were also given by Serre in his 1965 Harvard lecture notes, with a brief summary in his 1965 Algiers lectures (published by W.A. Benjamin and later translated). In the latter notes, he reverses the order of the key statements, but as far as I can tell one really needs Weyl's theorem first. (This issue comes up in a MO quetiuon here. The proof in section 6 of my 1972 text follows Bourbaki.)

2) But there is a largely separate line of work on linear algebraic groups, which owes even more to Chevalley and certainly merits the label "Jordan-Chevalley decomposition". Actually, a couple of papers by Kolchin in 1948 started in this direction, but Chevalley's 1951 book and his famous 1956-58 classification seminar made the results basic to all further work. The striking fact is that the semisimple and unipotent parts in the multiplicative Jordan decomposition are intrinsically defined in any connected linear algebraic group (over any algebraically closed field, though Chevalley's early work started out over arbitrary fields). Adaptations to fields of definition then follow.

It is this multiplicative version that Couty et al. emphasize in their work, which is partly historical and partly pedagogical.

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  • $\begingroup$ Thanks. The Couty et al. reference was helpful. The nicest proof I've seen of the Jordan decomposition "semisimple+nilpotent" is the one you give on page 17 of your 1972 text. I like that it is based only on the elementary idea of partial fractions (equivalently, Chinese Remainder Theorem). Do you know the history of that proof? $\endgroup$ Dec 19, 2013 at 20:31
  • $\begingroup$ @Drew: No, at this point I'm not sure exactly how that proof evolved. Certainly it wasn't due to me. When Couty contacted me a few years ago I was similarly unable to add to the historical picture she was reconstructing. $\endgroup$ Dec 19, 2013 at 23:54
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It seems that Hawkins' book on Frobenius answers many of my questions:

https://www.amazon.com/dp/1461463327/ref=olp_product_details?_encoding=UTF8&me=

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