Talk:Jordan normal form

Mischaracterization of diagonalizability?

The article states:

A n × n matrix A is diagonalizable if and only if the sum of the dimensions of the eigenspaces is n. Or, equivalently, if and only if A has n linearly independent eigenvectors.

But the 0 matrix is diagonalizable (indeed diagonal) without having this property. What am I missing?

a reply: the 0 matrix has that property. any vector is an eigenvector of the 0 matrix. Mct mht 12:37, 21 September 2006 (UTC)

But matrix ${\begin{bmatrix}1&0&0\\0&0&1\\0&0&0\\\end{bmatrix}}$ isn't diagonalizable (it's not similar to matrix with only one non zero entry) but it has two eigenspaces - one 1d and second 2d, which gives 3 dimentions. What am I missing? Micha7a (talk) 19:04, 1 July 2014 (UTC)

another reply: No contradiction. That example matrix has a 0-eigenspace with dimension 1 and a 1-eigenspace with dimension 1. So a set of independent eigenvectors for that 3 by 3 matrix has at most 2 elements, not diagonalizable. Akrodger (talk) 16:18, 10 December 2018 (UTC)

Source of Proof

Where was the proof adapted from? Source please. Also, are there any other sources for the real canonical form besides Shilov, Horn & Johnson, and Hirsch & Smale?—Preceding unsigned comment added by 71.202.54.148

I assume that this refers to the section that starts with "We give a proof by induction that any complex-valued matrix A may be put in Jordan normal form." Indeed that is a long section that was not tagged with sources. Someone also stuck a "disputed" template on it, but did not bother to post on this talk page what is being disputed.67.198.37.16 (talk) 23:20, 23 January 2019 (UTC)

The section "A proof" has several issues, for example:

1. The section begins with "We give a proof by induction." A proof of what? This should be stated in the section title or at the beginning.
2. The arguments seem too technical and difficult to understand.
3. The section cites no sources.
4. Due to items 2 and 3 above, the factual accuracy is difficult to verify.
5. The first paragraph is difficult to follow. For example, "The 1 × 1 case is trivial. Let A be an n × n matrix." If this paragraph describes the 1 × 1 case, then why refer to A as an n × n matrix? "Take any eigenvalue λ of A." If n = 1, why refer to "any eigenvalue λ of A? And so on.

I gave up reading after the first paragraph and inserted the multiple issues template. I think this section should be completely rewritten. If the proof were clear, it shouldn’t need any sources.—Anita5192 (talk) 07:19, 24 January 2019 (UTC)

Answers: 1. -- its a proof of the statement immediately before it, obviously. I quote directly: every square matrix A can be put in Jordan normal form is equivalent to the claim that there exists a basis consisting only of eigenvectors and generalized eigenvectors of A.

Answers: 2. -- what exactly is too technical? To me, it seems clear as a bell and is written at the same level of sophistication as the rest of the article. If you understood the rest of the article, I can't imagine why you wouldn't understand the proof.

Answer 3. -- In mathematics, the standard way of verifying factual accuracy is very definitely NOT to look it up somewhere, but determine if the proof is actually correct!

Answer 4. --It may seem difficult, but it is no harder than a homework problem or a class lecture. It doesn't use any concepts that didn't already appear earlier in the article.

Answer 5. -- then why refer to A as an n × n matrix? Because it is obviously a proof by induction. I mean, it tells you that it's a proof by induction, in the first sentence. All proofs by induction always start exactly in this way. The words "proof by induction" means that you prove two things: that the N=1 case holds, and that N implies N+1. If these two things hold, then it holds for all N. Since, for this case N=1 is trivial, the proof immediately jumps to general N. (Try it! Just plug in N=1 everywhere, and see what happens! By the end of the proof, you will have N=2. Now go back, plug in N=2...)

Answer 5. -- If n = 1, why refer to "any eigenvalue λ of A? Because matricies that are 1x1 are still matricies, and still have eigenvalues and eigenvectors!

I'm removing the prod. The proof appears to be sane; it appears to follow commonplace style and conventions for proofs. It appears to be written at the same level of sophistication as the rest of the article. It appeals only to entirely ordinary topics in linear algebra, like kernel, range, subspace and basis, topics that you would be expected to already know, before having your first encounter with Jordan normal forms. If you don't know what kernel, range, subspace and basis are, then ... pretty much none of this article is going to make sense. I don't see anything actionable, here. 67.198.37.16 (talk) 06:58, 5 February 2019 (UTC)

Thank you for cleaning up the section. It is much more readable now. When I inserted the "disputed" template on October 08, 2017, the section was difficult to understand. Since then, at least one other editor made some improvements, and now you have made it even better. —Anita5192 (talk) 07:40, 5 February 2019 (UTC)

I still think the proof section is far from clear. It is not clear to me what the claim is: 'every square matrix can be put into Jordan form' or 'a square matrix can be put into Jordan form if and only if there exists a basis consisting only of eigenvectors and generalized eigenvectors.' Those are two different claims. The paragraph immediately before the proof, under Generalized eigenvectors, does not justify the statements made. The paragraph preceding the proof, under Generalized eigenvectors, states facts without trying to explain why they are true. — Preceding unsigned comment added by 2601:546:C200:4CB0:8C57:7362:C5F8:4724 (talk) 00:59, 26 November 2019 (UTC)

Why is it relevant? How is it used?

— Preceding unsigned comment added by 199.83.40.35 (talk) 16:48, 27 March 2010 (UTC)

Something suspicious in section "Powers"

In section "Powers" what does the following matrix represent?

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

It is not a Jordan block because there are two eigenvalues on the diagonal; it's not a matrix in Jordan form because it's not a block matrix. Maybe the two eigenvalues should be the same?

Fixed. Algebraist 15:52, 25 June 2010 (UTC)

A more formal Jordan Chain?

Yes, they clearly exist, but there is no foundational explanation of what a Jordan chain is. I believe this may be a point of misunderstanding. Perhaps a small section explaining the existence of the Jordan chains without use of Jordan normal form would be enlightening? (As the notion of a Jordan chain is also used in the proof of the existence of a Jordan normal form) — Preceding unsigned comment added by 67.176.80.18 (talk) 20:13, 21 February 2013 (UTC)

sign mistake in part "Real matrices"?

Shouldn't $C_{i}={\begin{bmatrix}a_{i}&b_{i}\\-b_{i}&a_{i}\\\end{bmatrix}}$ be transposed since

${\begin{bmatrix}a&b\\-b&a\\\end{bmatrix}}{\begin{bmatrix}x\\y\\\end{bmatrix}}={\begin{bmatrix}ax+by\\-bx+ay\\\end{bmatrix}}$ ≠ ${\begin{bmatrix}ax-by\\bx+ay\\\end{bmatrix}}$

As would be required for the multiplication by λ=a+ib. This occurs in the part "Real matrices" — Preceding unsigned comment added by 137.73.8.8 (talk) 21:30, 21 March 2013 (UTC)

Yes, I believe that is correct; I just fixed it. 67.198.37.16 (talk) 02:15, 24 January 2019 (UTC)

Matrices with entries in a field

The page on Jordan-Chevalley decomposition has a counterexample. Where is the reference, anyway?--345Kai (talk) 12:01, 20 July 2016 (UTC)

Adding article to enhance to frobenius normal form and Weierstraß normal form

Dear math wiki folks, please mention the application of these and hint the limits of the form (and where to use)

Further the point how to enhance to that forms and that it is a generalization from polynomial depiction should be stated out. When using determinant entries equals 0 it is not feasible or needs to be enhanced.

Using group theory is nice, but there should be a short sum up to this as well for other folks. — Preceding unsigned comment added by 134.61.105.245 (talk) 14:47, 2 June 2017 (UTC)

Hmm. Frobenius normal form. Weierstraß normal form Weierstrass normal form, Weierstrass form. Applications. Phew. There are applications in atomic physics; assorted atomic and molecular spectra go degenerate like this in magnetic fields or whatever. I've forgotten the details. It shows up in engineering, something to do with degenerate spectra of vibrations. Shows up in shift spaces, with the Jordan chain being the shift; shift spaces show up in analysis of chaotic/ergodic motion. I've only the vaguest recollection of the details. Yes, examples should be given. 67.198.37.16 (talk) 02:23, 24 January 2019 (UTC)

Important not to round? What would be rounded anyway and why?

Does anybody know exactly what purpose the sentence, "As the next section explains, it is important to do the computation exactly instead of rounding the results." serves?

• Firstly, maybe I haven't looked closely enough, but I can't see an explanation of this in the 'next section'; and besides, this point should probably be worded better, with a specific reference to the section or even just a summary of the explanation if it's so important.
• "Important" to what? What will the side-effects be if there is rounding in the process?
• For that matter, why exactly might I be computing the Jordan normal form of a matrix of numbers either by hand or in some other situation where I might be tempted to round some of the results? I'm an algebraist, not an engineer, but surely either one is describing the Jordan normal form of some mostly-generic matrix, or a computer is performing the calculation already as precisely and accurately as it can? If there is some other application, or if a special command needs to be given to a computer for it to use sufficient precision when ordering it to convert matrices to Jordan normal, again I think this should be specified and probably even discussed.

I sincerely mean no offence, but is that sentence ridiculous as it is, or is it just me? Adam Dent (talk) 16:42, 5 April 2021 (UTC)

 Done I removed it for the reasons you mentioned. I also removed two other instances of the qualifier, "exactly," as it is superfluous in this article. Since this article is about linear algebra but not numerical linear algebra, it should be up to the user to decide whether and how to approximate. Thank you for pointing this out. —Anita5192 (talk) 16:57, 5 April 2021 (UTC)

Flat normal form

Is the content of the "flat normal form" section notable enough to be included here? It doesn't seem that way to me (and I am a professional mathematician). Especially with the name included explicitly in the text, it sounds suspiciously like a self-insert (or an insert by some misguided student). 185.49.31.254 (talk) 01:41, 30 April 2021 (UTC)