Talk:Kernel (matrix)

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Can somebody make a multi-dimesional example here? (more than one free variable....)

I am for the idea of merging this article (Null Space), with Kernel -- minghan 15:23, 26 January 2006 (UTC)[reply]

I would prefer to have kernel (mathematics) continue to focus on the more general case while this article addresses the restricted case of kernels of matrix operators in linear algebra. Deco 02:33, 27 January 2006 (UTC)[reply]

Changed my mind. This article presents itself as the same concept as kernels, so a merge is warranted. Deco 02:35, 27 January 2006 (UTC)[reply]

A slight philosophical difference is that a matrix is not a map, whereas a linear transformation is, even though the two are commonly identified. The null space of a matrix refers to what would be the kernel if we thought of it as a linear transformation. Whether the articles are written this way, I don't know, but I think that at least in theory, the two should be considered separate concepts, that you realize are the same after a little bit of thinking.

Hmmm. I still suggest that the merge discussion be centralized in kernel (mathematics). — Arthur Rubin | (talk) 02:57, 17 March 2006 (UTC)[reply]

Division by Zero

"Nullity" should no longer redirect here, since it appears that someone has managed to solve the "divide by zero problem"... and the solution is called Nullity. (http://www.bbc.co.uk/berkshire/content/articles/2006/12/06/divide_zero_feature.shtml) Medevilenemy 19:24, 6 December 2006 (UTC)[reply]

No offense, but that "solution" is total bullcrap. --Wooty Woot? contribs 09:54, 7 December 2006 (UTC)[reply]

It's not Wikipedia's job to decide whether a theory is correct or not. Leave that to discussion within the mathematics community and related journals. See the note at the bottom of WP:NOR. --Tjohns ✎ 12:03, 7 December 2006 (UTC)[reply]

I think discussion regarding this should take place over at Talk:Nullity. --Tjohns ✎ 12:00, 7 December 2006 (UTC)[reply]

Major revision

I just posted a major revision to this page, and I added the WikiProject Mathematics template above. Jim 03:06, 9 September 2007 (UTC)[reply]

That is not an "expansion" as your edit summary says, but a "limitation" to the special case of a finite dimensional linear operator. The definition is much wider than for matrices only. −Woodstone 06:05, 9 September 2007 (UTC)[reply]

I don't agree that the introduction to the article should focus on the general case like this. Here's why:

The null space of a matrix is an important idea, and most of the article is devoted to it. Null spaces of matrices are central to elementary linear algebra, and I think they deserve their own article.
Introducing null spaces in the context of general operators makes the article less accessible to a general audience.
For general linear operators, the null space is more often referred to as the kernel.

In an attempt to resolve this disagreement, I've created an article called kernel (linear algebra) that discusses the general case, and added a disambig template to the top of this article. Jim 18:45, 9 September 2007 (UTC)[reply]

You are missing the point again. Linear algebra is not the general case. Also non-linear operators can have a null space. Did you notice that there are already quite many "kernel (something)" articles? Why add another one. In my opinion the titles should be reversed from your idea. Null space is generic. The "Matrix" case is quite specific and should have an appropriate title if you insist on splitting it from the generic one. −Woodstone 07:09, 10 September 2007 (UTC)[reply]

I'm not sure I've ever heard of anyone talk about the null space of a nonlinear operator. Most of the time people use the word "operator" to mean "linear operator", and even in that case the "null space" is most commonly called the "kernel". My experience has been that the word "null space" is used most of the time to talk about the null space of a matrix.

Google backs me up on this. There are 476,000 results for "null space", of which 347,000 involve the word "matrix", and 413,000 involve either the word "matrix" or the word "linear". The remaining hits don't particularly seem to be about the null space of a nonlinear operator.

The Math World article on null spaces restricts to "linear transformations", as does the Merriam-Webster definition. I looked at the Wikipedia articles that link to "Null space", and I didn't notice any that have to do with nonlinear operators.

I like the current organization of the articles on null space and kernel (linear algebra), but if you strongly object to then we could describe our disagreement on Wikipedia talk:WikiProject Mathematics, and ask what the other members think. Jim 17:17, 10 September 2007 (UTC)[reply]

Since you've reverted the introduction again, I've asked for an outside opinion from Wikipedia talk:WikiProject Mathematics. (Those coming from the outside should take a look at the current version and previous version of the introduction.) Jim 01:18, 14 September 2007 (UTC)[reply]

I don't believe the set of solutions of

f(x)=0

for a general f is called null space. It's only called a null space if f is linear. I think it's called a kernel if f preserves the relevant algebraic structure, and the expression for the general situation is zero set.

I'm not so sure about having separate articles on null space (of matrices) and kernel (linear algebra) (of linear operators). There is a big overlap and I'm not convinced that the article would be less accessible if we were to treat linear operators here, provided that we do introduce null spaces for matrices first and that we delineate the more abstract parts which rely on identifying matrices with linear operators. If we do decide to have separate articles, then I think they should be linked more than by just a note at the top. -- Jitse Niesen (talk) 02:24, 14 September 2007 (UTC)[reply]

The introduction in terms of a matrix looks better to me, more accessible. It's OK to have a separate matrix-oriented article, and a separate operator-oriented article, with more of a functional analysis and algebra flavor (not the college lower division linear algebra). These really serve distinct audiences. I am not sure if the kernel vs nullspace nomenclature has anything to do with this distinction - they are exactly the same to me - so perhaps some renaming is in order. And, yes, I agree with Jitse Niesen that both refer to linear operators only. By the way, computing the basis of the nullspace of a matrix by means of elementary transformation as done in the article now and in undergraduate textbooks is unfortunate and of (perhaps) a didactic value only because of numerical stability problems; practical computation should be done via QR decomposition or better SVD, like in Matlab function null. Jmath666 04:34, 14 September 2007 (UTC)[reply]

I agree that there's a problem with presenting only the row reduction perspective, and I've been trying to figure out how to deal this problem for many different linear algebra articles (c.f. column space, row space, Euclidean subspace, and system of linear equations.) On the one hand, row reduction is the standard algorithm given in most introductions to linear algebra; on the other hand, it apparently has serious numerical stability problems, making it relatively useless for practical applications. See Talk:System of linear equations for more discussion about this issue. Jim 20:45, 14 September 2007 (UTC)[reply]

I did not say there is a problem with row reduction as opposed to columns; there is a problem with the reduction approach at all. This results in unfortunate engineers who try to implement those naive methods out of the book in software. But the educational establishment deems it important so students are subjected to it. Hence it has a place on Wikipedia where those students may look. If I have time I may add a paragraph here about computing the nullspace in a numerically sound manner. Yes the issues in solving systems are similar, but simpler. See also the paragraph on numerical computation in Moore-Penrose pseudoinverse. For example Trefethen-Bau ISBN 978-0898713619 do not even consider Gaussian elimination (a.k.a. reduction) for anything and start with QR. Matlab switched from QR to SVD (more expensive but more reliable) for things like the nullspace many years ago; reduction methods are not even a remote consideration for many decades now if reliability is important. Jmath666 04:48, 15 September 2007 (UTC)[reply]

Sorry, we seem to be miscommunicating. I was agreeing with you—by "row reduction" I mean Gaussian elimination, or any method that involves elementary row and/or column operations. I'm calling it "row reduction" just because it's usually done with rows in introductory courses. I know about the numerical problems with reduction, and I've been trying to figure out how this should be reflected in the articles on elementary linear algebra. See my last post. Jim 09:07, 15 September 2007 (UTC)[reply]

I'm not sure we need a separate article kernel (linear algebra) next to kernel (algebra)#Linear operators, but if kept, there should be cross references. --Lambiam 08:58, 14 September 2007 (UTC)[reply]

Even if I understand that matrices are more common use, nullspace does not only concern finite dimensional spaces, and not only real vector spaces (in particular, complex matrices and matrices over finite fields are also widely used). Hence the definition, at least, should not restrict the term to such a particular situation. Moreover, in the article, the term matrix is used in place of "real matrix", leading to strange assertions like "The null space of an m × n matrix is a subspace of Rn".Also,the introduction of the term "Euclidean space" is misleading: Why should the natural metrics be taken into account? This is particularly funny if A is used to define a quadratic form associated to a non-Euclidean distance. Should we consider that the isotropic subspace of a quadratic form is an Eucldiean subspace??? pom 21:33, 18 September 2007 (UTC)[reply]

Renaming proposed

In view of the discussion above I propose there should be

kernel (linear algebra), including elementary linear algebra and numerical linear algebra aspects (this article)
kernel (functional analysis), or operator theory (relating/including Fredholm theory etc)
kernel (algebra) (as is now, perhaps incorporating kernel (operator theory))

with proper redirects from the matching variants of nullspace, and a dab page. Each with distinct audience and at a distinct level. For example, referring to the post by Taxipom, the words "natural metric" and all considerations that follow will make no sense whatsoever to a typical 2nd year US college student who just takes a first course in linear algebra and wants to use wikipeda to look up something. While it would be all fine and of interest to a grown mathematician or a student in a classical, pure math oriented program. Or... maybe I'll just be bold do it and see if anyone objects. Jmath666 04:12, 19 September 2007 (UTC)[reply]