# Wikipedia:WikiProject Mathematics/PlanetMath Exchange/15-XX Linear and multilinear algebra; matrix theory

This page provides a list of all articles available at PlanetMath in the following topic:

15-XX Linear and multilinear algebra; matrix theory.

This list will be periodically updated. Each entry in the list has three fields:

1. PM : The first field is the link to the PlanetMath article, along with the article's object ID.
2. WP : The second field is either a "guessed" link to a correspondingly named Wikipedia article, produced by the script which generated the list, or one or more manually entered links to the corresponding Wikipedia articles on the subject.
3. Status : The third field is the status field, which explains the current status of the entry. The recommended status entries are:
 Status means PM article N not needed A adequately covered C copied M merged NC needs copying NM needs merging
• Please update the WP and Status fields as appropriate.
• if the WP field is correct please remove the qualifier "guess".
• If the corresponding Wikipedia article exists, but the link to it is wrong, please fix the link.
• If you copy or merge an article from PlanetMath, please update the WP and Status fields for that entry.
• If you have any comments, for example, thoughts on how the PlanetMath article compares to the corresponding Wikipedia article(s), please place such comments on a new indented line following the entry. Comments of this kind are very valuable.

Don't forget to include the relevant template if you copy over text or feel like an external link is warranted

• {{planetmath|id=|title=}} for copied over text
• {{planetmath reference|id=|title=}} for an external link

See the main page for examples and usage criteria.

One can use the web-based program Pmform to convert PlanetMath articles to the Wikipedia format. As a side benefit, this tool will place the PlanetMath template for you.

## 15-00 General reference works (handbooks, dictionaries, bibliographies, etc.)

Copied and made suitable for WP. Jitse Niesen (talk) 13:58, 24 July 2005 (UTC)
The PM article has some topological information on the antipodal map, which is now merged into the WP article. Jitse Niesen 21:38, 19 Mar 2005 (UTC)
The WP article had most of the information of the PM article and quite a bit more. I merged the remaining bits of the PM article into the WP article and rewrote it in the process. Jitse Niesen 21:38, 19 Mar 2005 (UTC)
The WP article lacked some alternative names and symbols and some basic properties, which are now included. Note that the related article Hermitian adjoint could use some clean-up. Jitse Niesen 15:49, 22 Mar 2005 (UTC)
The corollary is that any normal matrix is unitarily similar to a diagonal matrix. This is mentioned in Schur decomposition and spectral theorem. The PM article includes a proof which we don't need. Jitse Niesen 15:49, 22 Mar 2005 (UTC)
PM article is two lines and only mentioned the definition. I created a redirect to dual space, where the name covector is mentioned. Jitse Niesen 15:49, 22 Mar 2005 (UTC)
PM article states that if f is homogeneous of degree r, then the derivative of f is homogeneous of degree r − 1 (it also includes a proof which we do not need). This should be mentioned in an article on homogeneous functions, which unfortunately does not exist (either that or I could not find it). See also homogeneous (mathematics). Jitse Niesen 17:37, 24 Mar 2005 (UTC)
Two sentences were merged into the WP article, which was largely complete. Jitse Niesen 17:37, 24 Mar 2005 (UTC)
PM article only defines what diagonally dominant is. I think that is not enough to warrant an article. Jitse Niesen 23:20, 24 Mar 2005 (UTC)
I created the article today. Diagonal dominance has a number of important uses in numerical linear algebra (tho' I only started a list). Could someone add the stub tag? And is there a category for numerical linear algebra? (Or just linear algebra and numerical analysis separately?) Thanks.Lunch 18:50, 23 May 2006 (UTC)
Phew ... this was a lot of work since the WP article was in a rather bad state, although it was largely complete. Now, the WP article is completely rewritten and some statements from the PM article are merged in. Jitse Niesen 22:17, 19 Apr 2005 (UTC)
This is a tough one. The PM article is a wide-ranging discussion. Synopsis of PM article (with corresponding WP articles): definition (eigenvalue, eigenvector), differential eigenvalue problems (differential operator, spectrum of an operator, Sturm-Liouville problem), matrix eigenvalue problems (again eigenvalue), numerical eigenvalue problems (eigenvalue algorithm, QR algorithm). I don't know what to do with this; I think we should merge in some material in the mentioned WP-articles and leave it with that; I already expanded the WP article on the QR algorithm. Jitse Niesen 22:17, 19 Apr 2005 (UTC)
Update: The new article eigenvalue, eigenvector and eigenspace has similar scope as the PM article.
PM article is a longwinded (IMHO) proof that all eigenvalues of orthogonal matrices have unit modulus; a fact which can be more easily proved from the isometry property. I added a statement to that effect. Jitse Niesen 22:17, 19 Apr 2005 (UTC)
PM article contains a trivial proof of the form of 2 × 2 equitable matrices. Jitse Niesen 21:19, 25 Mar 2005 (UTC)
A matrix is equitable if it satisfies $a_{ij}=a_{ik} a_{kj}$. I doubt this concept is notable enough to be included in WP. Indeed, a Google search turned up only two references to the term "equitable matrix" and I could find only one article in MathSciNet (Parker, Matrices in the market place, Math. Mag. 38:125--128, 1965), describing an application in economics also listed on PM. On the other hand, PM also has a reference to "Elementary matrix theory" by H. Eves. Jitse Niesen 12:37, 4 Apr 2005 (UTC)
Entire content of PM article: "An exactly determined system of linear equations has precisely as many unknowns as equations and is hence soluble." Jitse Niesen 21:19, 25 Mar 2005 (UTC)
I now mentioned the concept exactly determined at system of linear equations. -- Jitse Niesen (talk) 23:18, 1 September 2005 (UTC)
PM article explains the construction of a free vector space. As far as I can tell, this construction also works for modules, so I merged it into the WP article on free modules. Jitse Niesen 17:29, 21 Apr 2005 (UTC)
In fact, the PM and WP articles are very similar. I only added a URL listed in the PM article as a reference to the WP article and did some copy-editing. Jitse Niesen 17:29, 21 Apr 2005 (UTC)
I created the WP article homogeneous function by copying the relevant parts of the disambiguation page homogeneous. Jitse Niesen 00:01, 22 Apr 2005 (UTC)
PM article is a rather indirect proof of the nilpotency of strictly triangular matrices. Jitse Niesen 23:48, 22 Apr 2005 (UTC)
It would be nice to have an example in substitution rule for the case of multivariate integrals, but I don't like to PM example very much. Jitse Niesen 23:48, 22 Apr 2005 (UTC)
The theorem was mentioned on WP, but not the fact that it is called "Jacobi's theorem". Jitse Niesen 23:48, 22 Apr 2005 (UTC)
I spun off parts of tensor product and matrix multiplication and added material from the PM article and some more. Jitse Niesen 22:00, 8 Jun 2005 (UTC)
Unfortunately, the PM article, which is copied from another source, is rather bad. Jitse Niesen 22:00, 8 Jun 2005 (UTC)
I actually did this some time ago, making a terrible mistake, fortunately caught by Michael Hardy. Jitse Niesen (talk) 23:01, 19 July 2005 (UTC)
WP article has a good elementary intro but degenerates after that (e.g., it lists a dozen equivalent statements for "A is invertible"), but the PM article does not treat any advanced parts. In short, WP article needs a serious overhaul. Jitse Niesen (talk) 23:01, 19 July 2005 (UTC)
Jitse Niesen (talk) 23:01, 19 July 2005 (UTC)
A linear manifold is just another name for an affine space, which is treated briefly but adequately at affine transformation. I created a redirect from linear manifold. Jitse Niesen (talk) 23:01, 19 July 2005 (UTC)
Jitse Niesen (talk) 13:58, 24 July 2005 (UTC)
Our list is at least twice as long, but still, PM had two entries which we did not have: Anti-diagonal matrix and Pascal matrix. Jitse Niesen (talk) 13:58, 24 July 2005 (UTC)
But I could still add some more to the WP article. -- Jitse Niesen (talk) 22:25, 6 August 2005 (UTC)
I added the redirect matrix operationmatrix (mathematics). -- Jitse Niesen (talk) 22:25, 6 August 2005 (UTC)
Jitse Niesen (talk) 22:25, 6 August 2005 (UTC)
Almost done merging, only the section on flags needs to be copied. -- Jitse Niesen (talk) 22:25, 6 August 2005 (UTC)
Now done. -- Jitse Niesen (talk) 22:33, 10 August 2005 (UTC)
Entire content of PM article is "A non-zero vector in a vector space V is a vector that is not equal to the zero vector in V." which seems all there is to say and would end up at VfD here. -- Jitse Niesen (talk) 22:25, 6 August 2005 (UTC)
I later found out that zero vector is a redirect to null vector (vector space), so non-zero vector also redirects there now. -- Jitse Niesen (talk) 22:33, 10 August 2005 (UTC)
Entire content of PM article is "Let $A=(a_{ij})$ be a square matrix. An element $a_{ij}$ is an off-diagonal entry if $a_{ij}$ is not on the diagonal, i.e., if ij." -- Jitse Niesen (talk) 22:33, 10 August 2005 (UTC)
PM had the interesting fact that orthogonal matrices, especially Householder reflections and Givens rotations, are important in numerical analysis. For the rest, the WP article is far more complete. -- Jitse Niesen (talk) 23:18, 1 September 2005 (UTC)
Jitse Niesen (talk) 23:18, 1 September 2005 (UTC)
I just mentioned the word overdetermined in system of linear equations. -- Jitse Niesen (talk) 23:18, 1 September 2005 (UTC)
Jitse Niesen (talk) 00:56, 4 September 2005 (UTC)
I copied the inequality, but left out the proof. -- Jitse Niesen (talk) 00:56, 4 September 2005 (UTC)
Jitse Niesen (talk) 23:10, 6 September 2005 (UTC)
WP article is longer and covers all the main points of the PM article. Jitse Niesen (talk) 23:12, 9 September 2005 (UTC)
WP already has a different proof. Jitse Niesen (talk) 23:12, 9 September 2005 (UTC)
This is a constructive proof which is used in numerical linear algebra. Jitse Niesen (talk) 17:56, 8 October 2005 (UTC)
WP article is clearly more complete. Jitse Niesen (talk) 23:12, 9 September 2005 (UTC)
While the Wikipedia article was similar to the PM article, it did need to be explanded a bit, inter alia with the proof mentioned two items above. Jitse Niesen (talk) 17:56, 8 October 2005 (UTC)
The PM article notes that the SVD can be used to solve least-squares problems. I copied this statement. Jitse Niesen (talk) 17:56, 8 October 2005 (UTC)
I merged properties 3 and 4 from the PM article into our article. Jitse Niesen (talk) 17:56, 8 October 2005 (UTC)
WP article is more complete. Jitse Niesen (talk) 22:48, 5 October 2005 (UTC)
The PM article only contains the definition, which was already present in the WP article. Just to be sure, I redirected strictly upper triangular matrix and strictly upper triangular matrix to triangular matrix. -- Jitse Niesen (talk) 17:18, 10 October 2005 (UTC)
PM article only contains the definition, but WP article has more. -- Jitse Niesen (talk) 17:18, 10 October 2005 (UTC)
The theorem is that every normal triangular matrix is diagonal. I added this to the WP articles, with the proof condensed to one line. -- Jitse Niesen (talk) 17:18, 10 October 2005 (UTC)
PM article included the fact that the eigenvalues of a triangular matrix are the diagonal entries; I added this to the WP article which was in all other respects more complete. -- Jitse Niesen (talk) 17:18, 10 October 2005 (UTC)
PM only contains the definition, and WP quite a bit more. -- Jitse Niesen (talk) 14:39, 5 November 2005 (UTC)
I made redirects from Under determined and Underdetermined to System of linear equations. -- Jitse Niesen (talk) 14:39, 5 November 2005 (UTC)
I created a redirect from Unit triangular matrix to Triangular matrix, where this concept is mentioned. -- Jitse Niesen (talk) 14:39, 5 November 2005 (UTC)
PM says that a unitary space is a complex vector space with a distinguished positive definite Hermitian form, which serves as inner product. I am not sure we need this definition here. The other concepts treated by PM are adequately covered in Unitary operator and Unitary matrix. -- Jitse Niesen (talk) 14:39, 5 November 2005 (UTC)
PM article mentioned the sum of two subspaces, and the orthogonal complement. I added both to WP. -- Jitse Niesen (talk) 19:42, 5 November 2005 (UTC)
Redirect created, analogous to zero function which also redirects to 0 (number). -- Jitse Niesen (talk) 19:42, 5 November 2005 (UTC)
Utter triviality, but I created the redirect nonetheless. -- Jitse Niesen (talk) 19:42, 5 November 2005 (UTC)
• PM: point, id=8173new! -- WP guess: point -- Status:

## 15-01 Instructional exposition (textbooks, tutorial papers, etc.)

The PM article does have a more general definition, but I have my doubts about it (what if A = {1,2,3,4} and σ = (1 3)(2 4)? ), and there is no reference. -- Jitse Niesen (talk) 23:22, 22 November 2005 (UTC)
Added the formula to the WP article, which could use some rigorous editing. -- Jitse Niesen (talk) 00:34, 23 November 2005 (UTC)
Merged some (rather trivial) information about eigenvalues and eigenvectors. -- Jitse Niesen (talk) 00:34, 23 November 2005 (UTC)
Lots of information in both PM and WP article. Articles clearly differ in what information is listed, with WP one being clearly more low-level, but both articles are essentially fine. -- Jitse Niesen (talk) 21:28, 29 November 2005 (UTC)
PM article has some more things, but it's all trivial. -- Jitse Niesen (talk) 21:28, 29 November 2005 (UTC)

## 15A03 Vector spaces, linear dependence, rank

Jitse Niesen (talk) 21:28, 29 November 2005 (UTC)
I mentioned the term "complementary subspace" in Direct sum of modules. It might be good to make Direct sum of vector spaces in a separate article, because the module stuff is a bit more abstract. -- Jitse Niesen (talk) 15:27, 18 December 2005 (UTC)
This definition was hinted at in various places, but I think it can use an article on its own. -- Jitse Niesen (talk) 15:27, 18 December 2005 (UTC)
I don't think we need an article for this; see diagonalizable matrix and quadratic form. -- Jitse Niesen (talk) 01:27, 20 December 2005 (UTC)
WP article is actually more complete. -- Jitse Niesen (talk) 01:27, 20 December 2005 (UTC)
PM article says basically that every matrix can be written as a sum of a Hermitian and a skew-Hermitian matrix, which is already mentioned in the two WP articles. I don't think we need to make more of it . -- Jitse Niesen (talk) 01:27, 20 December 2005 (UTC)
WP article is perfectly adequate. -- Jitse Niesen (talk) 01:27, 20 December 2005 (UTC)
PM includes a proof, using Zorn's lemma, but I don't think we need that. -- Jitse Niesen (talk) 01:27, 20 December 2005 (UTC)

### I am here

• PM: flag, id=2994 -- WP guess: flag -- Status:
• PM: frame, id=2931 -- WP guess: frame -- Status:
Actually, I just made the WP article redirect to null space, which has all the info on PlanetMath and more. Oleg Alexandrov 17:57, 16 May 2005 (UTC)
• PM: rank, id=2236 -- WP guess: rank -- Status:
• PM: span, id=806 -- WP guess: span -- Status:

## 15A04 Linear transformations, semilinear transformations

• PM: trace, id=1844 -- WP guess: trace -- Status: