Talk:Rank (linear algebra)

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rk(A)

I was rather surprised by the statement in the article that the rank "is usually denoted rk(A)". So I checked some books, with the following result:

Most books actually do not introduce a notation for the rank of a matrix.
Five books use rank A, namely Linear Algebra and Geometry by Bloom, Topics in Matrix Analysis by Horn and Johnson, Linear Algebra by Friedberg et al., Linear Algebra by Satiste, and Berkeley Problems in Mathematics by De Souza and Silv.
Three books use rk(A), namely Elements of Linear Algebra by Cohn, Linear Algebra by Jänich, and Linear Algebra by Kaye and Wilson.
Two books use r(A), namely Linear Programming by Hartley, and Linear Algebra with Applications by Scheich.

I changed the article accordingly. -- Jitse Niesen 23:44, 21 Aug 2003 (UTC)

wow there is only seven books in existence on this subject? ;) --LeakeyJee (talk) 13:14, 5 June 2008 (UTC)[reply]

Rank of the product of two matrices

The Rank (linear algebra) page states:

If B is an n-by-k matrix with rank n, then AB has the same rank as A.
If C is an l-by-m matrix with rank m, then CA has the same rank as A.

Does anyone have a proof (or reference to a proof) for this? Maybe it's obvious and I'm just not seeing it. Connelly 15:49, 7 September 2005 (UTC)[reply]

It's not that obvious. Sketch of the proof: Think of the matrices as linear transformations. If B is an n-by-k matrix with rank n, then the function x |-> Bx is surjective, hence the range of the function x |-> ABx is the same as the range of the function x |-> Ax, hence the ranks are equal. I'll see whether I can find a reference (rectangular matrices always confuse me). Let me know if you want me to elaborate. PS: Thanks for your edit to Hermitian matrix. -- Jitse Niesen (talk) 16:25, 7 September 2005 (UTC)[reply]

It follows from (0.4.5c) in Horn & Johnson, Matrix Analysis, which states (without proof): If A is m-by-n and B is n-by-k then

\operatorname {rank} \,A+\operatorname {rank} \,B-n\leq \operatorname {rank} \,AB\leq \min\{\operatorname {rank} \,A,\operatorname {rank} \,B\}.

If rank B = n, then this becomes rank A ≤ rank AB ≤ rank A. -- Jitse Niesen (talk) 19:49, 7 September 2005 (UTC)[reply]

Wow, thanks! I didn't expect a response so soon. Your proof works for me, but I'll check out the Matrix Analysis book too. I'm actually trying to show a more complex result, but I needed to check the validity of the Wikipedia statement first. I can post up your linear transformation proof on Wikipedia if you think that's a good idea (not really sure where to put it...maybe the Rank page or as a separate page linked to from Rank?). - Connelly 23:35, 7 September 2005 (UTC)[reply]

I think the proof would make a nice addition if it's kept short, because it explains the concept of rank and how to handle it. It's probably more important to mention the double inequality for rank AB (by the way, how hard would it be to prove that?). Generally, proofs on Wikipedia are a contentious issue and need to be considered on a case-by-case basis (how important is the proof and how much does it disrupt the flow of the article?). You can read a discussion about it on Wikipedia:WikiProject Mathematics/Proofs, which also has a proposal for putting proofs on a separate page. -- Jitse Niesen (talk) 12:16, 8 September 2005 (UTC)[reply]

There is a proof at http://books.google.com/books?id=5U6loPxlvQkC&printsec=frontcover at page 95. --Dagcilibili (talk) 18:50, 2 December 2008 (UTC)[reply]

Matrix rank definition with minor

Another definition of matrix rank:

The matrix A has rank r if it has a minor of size r which is different from zero and every minor of size r + 1 is equal to zero.

Ring question

The article says There are different generalisations of the concept of rank to matrices over arbitrary rings. In those generalisations, column rank, row rank, dimension of column space and dimension of row space of a matrix may be different from the others or may not exist. It doesn't distinguish between rings and commutative rings. Is it true that the generalisation to just commutative rings also has all of these issues? (it seems likely, and if it is true I think it would be useful to mention it since it would make the statement much stronger) A5 18:28, 19 March 2006 (UTC)[reply]

I realise this request is quite old now, but I also think that some more information on the definitions of rank for different classes of rings would be helpful. In my case, this is motivated by a need for information on which of these definitions cease to be equivalent for commutative rings with 1. —Preceding unsigned comment added by 128.40.159.177 (talk) 17:24, 15 July 2008 (UTC)[reply]

How about "rank deficient"

I think the page should mention the term "rank deficient". MusicScience 23:32, 12 January 2007 (UTC)[reply]

Term sounds familiar, and has the benefit of being self-explanatory. However, a quick Google search for 'intitle:"matrix algebra" "rank deficient"' finds only 7 distinct websites. Sources? References? Textbooks? -- JEBrown87544 18:19, 17 January 2007 (UTC)[reply]

--it would be helpful to show the relationship between rank and row space. -EH — Preceding unsigned comment added by 128.135.96.110 (talk)

Relation to condition number?

It seems like the rank is related to the condition number in that the rank describes the number of nonzero eigenvalues whereas the condition number describes the range of those eigenvalues. For practical purposes, if the eigenvalues, normalized by the largest, are (1.0, 0.9, 0.5, 1e-5, 1e-6), the rank is effectively three. Is there a page about this idea, relaxing the definition of "rank" to mean "very small eigenvalue"? Should condition number be in the see-also list on this page? —Ben FrantzDale (talk) 19:18, 20 October 2008 (UTC)[reply]

False statement

In the article it states that the rank of A is equal to the rank of A^T.A. This is clearly false. Take for example the column matrix (1 \\ 1) over the field of two elements. Then A^T.A = (0). It's true over the reals, and the given proof only works over the reals. - Dave Benson, Aberdeen, 15 Mar 2009 —Preceding unsigned comment added by 86.166.25.60 (talk) 08:15, 15 March 2009 (UTC)[reply]

Right. That list probably includes more properties that are only true over the reals or complex numbers. However, I don't have intuition for matrices over other fields, so I did'nt check them all. To be sure I added a bit at the start saying that we assume that the field is R or C. -- Jitse Niesen (talk) 14:08, 15 March 2009 (UTC)[reply]

I've just done a substantial rewrite of the article; in particular, I think that I've checked all of the listed properties and added the qualifier to the real or complex numbers wherever it was needed. --JBL (talk) 16:33, 8 March 2013 (UTC)[reply]

Is the definition correct ?

Since the column rank and the row rank are always equal, they are simply called the rank of A. How can they always be equal?

They are always equal, it is indeed amazing. — Preceding unsigned comment added by 69.174.157.147 (talk) 04:16, 14 November 2020 (UTC)[reply]

Row and column operations example

Right now there's an elaborate example using row operations. The LaTeX in this example is not very clear and the notations used are not defined in the article and are not standard. Any suggestions for how to rewrite the example are welcome. --JBL (talk) 20:45, 17 March 2013 (UTC)[reply]

I've noticed that the negative sign in front of 2 , 3 in row 2 are missing, but I see them when editing. Any idea why? — Preceding unsigned comment added by 91.18.252.46 (talk) 10:01, 22 January 2021 (UTC)[reply]

It seems that ut is a bug of some browsers. Try changing the zoom factor of your display. D.Lazard (talk) 10:29, 22 January 2021 (UTC)[reply]

Third proof

Why the third proof is commented out? — Preceding unsigned comment added by Raffamaiden (talk • contribs) 14:04, 18 May 2013 (UTC)[reply]

The third proof is limited to the case of real matrices. The proof is substantially a proof of a different result, using much more technology than is necessary or sensible to prove one of the first elementary results in linear algebra. So I removed it. JBL (talk) 15:32, 18 May 2013 (UTC)[reply]

JBL, that's absurd. The third proof is entirely equivalent to the second proof, and it conveys the same information more efficiently. I disagree with your assessment that the use of "technology" here is non-sensible. I am putting it back in. Bengski68 (talk) 15:21, 4 June 2020 (UTC)[reply]

@Bengski68: since you did not ping me, I've just seen this now. What is absurd is to invoke the idea of a Euclidean norm in a complex space in order to prove row rank and column rank are equal (a fact that is true over any field and does not depend in any way on the existence of a norm). Also "it's entirely equivalent to the second proof" is a good reason to remove it, not a good reason to keep it. -JBL (talk) 16:24, 14 November 2020 (UTC)[reply]

To editors JayBeeEll and Bengski68: I agree with JBL that only proofs that are valid on any field must be given. However, I wonder that the simplest and the most useful proof is not mentioned in the section on proofs, although it has been given in the preceding section (use of Gaussian elimination). I'll edit the article for filling this gap. IMO, the given proofs are more confusing than useful, as they are more technical and involve less elementary. I suggest to remove them. D.Lazard (talk) 17:15, 14 November 2020 (UTC)[reply]

What about numerical rank

Why does numerical rank go unmentioned on Wikipedia? If A≈BC are matrices and B is a n-by-k matrix, and C is a k-by-m matrix. Then the inner dimension k is often called the numerical rank of A. — Preceding unsigned comment added by 145.107.106.82 (talk) 14:35, 3 July 2017 (UTC)[reply]

@145.107.106.82:: see the section alternative definitions, where the definition you ask about is discussed under the name "decomposition rank", with a link to the article rank factorization. Do you have a reference for the term "numerical rank"? If so we could add it. --JBL (talk) 14:50, 3 July 2017 (UTC)[reply]

Relationship with eigenvalues

I am wondering why the relationship between the rank of a matrix and its eigenvalues/vectors not mentioned in this page. Manoguru (talk) 22:23, 29 November 2018 (UTC)[reply]

"Rank deficiency" listed at Redirects for discussion

A discussion is taking place to address the redirect Rank deficiency. The discussion will occur at Wikipedia:Redirects for discussion/Log/2020 July 17#Rank deficiency until a consensus is reached, and readers of this page are welcome to contribute to the discussion. 1234qwer1234qwer4 (talk) 17:50, 17 July 2020 (UTC)[reply]

Sum of 1-rank matrices (equivalent definition)

rank $A\leq r\ \Leftrightarrow \ A=\sum _{k=1}^{r}c_{k}v_{k}^{T}$ for some $c_{k}\in F^{m},\ v_{k}\in F^{n}$ . If rank $A=r$ , you can have the vectors $c_{k}$ equal to any independent columns of $A$ .

That should be added in the section on equivalent definitions. Likely also the row equivalent should be mentioned. --Rigmat (talk) 09:38, 11 May 2022 (UTC)[reply]

This is item 3 in § Decomposition rank. The second sentence is item 2 or 4. D.Lazard (talk) 10:04, 11 May 2022 (UTC)[reply]