Talk:Jordan normal form

message: 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)
replies: text:  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)
replies: text:  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)
thread_title:  Mischaracterization of diagonalizability? 

message: — Preceding unsigned comment added by 199.83.40.35 (talk) 16:48, 27 March 2010 (UTC)
thread_title:  Why is it relevant? How is it used? 

message: 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)
thread_title:  Something suspicious in section "Powers" 

message: 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)
thread_title:  A more formal Jordan Chain? 

message: 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)
replies: text: Yes, I believe that is correct; I just fixed it. 67.198.37.16 (talk) 02:15, 24 January 2019 (UTC)
thread_title:  sign mistake in part "Real matrices"? 

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

message: 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)
replies: text: 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)
thread_title:  Adding article to enhance to frobenius normal form and Weierstraß normal form 

message: 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)
replies: text: 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)
thread_title:  Important not to round? What would be rounded anyway and why? 

message: 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)
thread_title:  Flat normal form