Talk:Projection (linear algebra)

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I have seen the word projection used for this, but never projector, until today. Does someone use the former for linear transformations and the latter for matrices? If so, it should say so. Or do people in some scientific fields use one word while others use the other? Michael Hardy 20:09, 17 Apr 2004 (UTC)

I get the word projector for this from the book Numerical Linear Algebra by Lloyd N. Trefethen and David Bau, ISBN 0-89871-361-7. Here it is used to describe the linear transformation, whether or not its matrix is important, although in defining a projector, they unavoidably do use a matrix definition. I honestly don't know the difference between how the two terms are used, though it would make sense that in the context of numerical linear algebra, where algorithms are more important, the word projector, which more clearly conveys the action "to project" would be the more attractive choice. CyborgTosser

Is this talking about the same thing? Projection_operator

This is a terrible article. I really hope someone with some spare time and a better understanding of what people are looking for when they search for "projection" will edit this. Myrkkyhammas

Contents 1 "technical" and "context" tags 2 Affine case 3 First sentence 4 abstract example makes no sense 5 Projection operator 6 eigenvalue proof 7 Properties Section 8 Non-linear subspaces 9 Orthogonal Projection Section

[edit] "technical" and "context" tags

i removed these tags. i added a sentence to the intro about graphical projection and added an "other uses" tag to the top of the article. between that and the title of the article (which specifies that this is the use in linear algebra), is that enough? Lunch 22:57, 12 September 2006 (UTC)

Adding the other uses of projection was good idea. I think the technical tag may be resulting from the section with the example of orthogonalization. It was the only part of the article that I had to really think about what I was reading to understand. Archmagusrm 14:54, 2 October 2006 (UTC)

I disagree that this article doesn't need these tags, it's very obscure. Pdbailey 03:10, 24 October 2006 (UTC)

I think I agree with you (if I follow your double negative correctly). This article needs work. I removed the tags because the person who originally left them gave no comments about what they didn't understand or suggestions as to how the article could be improved.

Thanks for editing the intro. It has provided a needed push.

I think the old example that was there was frickin' ugly. The new example that was added was a bit too simple, but does let people know right off the bat what's going on. An intermediate example is needed. Maybe something along the lines of the rank-1 matrix described in the abstract section but more concrete. Perhaps an infinite dimensional example would be nice too.

The range and domain of the projection need to be connected back to the eigendecomposition of the projection. (And it could be noted that a projection always has a complete set of eigenvectors.)

Sigh. So much to do. Lunch 21:58, 24 October 2006 (UTC)

[edit] Affine case

Should this article include affine-linear projections? Or maybe they are covered somewhere else? An example of what I mean in 2 dimensions is P((x,y)) = (2,y), which is a projection onto the vertical line through (2,0). McKay 16:07, 18 October 2006 (UTC)

I don't think so, because this is an article about linear algrebra. Pdbailey 03:10, 24 October 2006 (UTC)

Affine transformations are a standard part of linear algebra. McKay 01:31, 25 October 2006 (UTC)

you could add a sentence about it to the effect that affine projections can be expressed as the composition of a translation and a projection. i'd keep it brief and not say much more than that though. Lunch 02:28, 25 October 2006 (UTC)

[edit] First sentence

Currently, the current first sentence is

In linear algebra, a projection is a linear transformation P from a vector space to itself such that P² = P.

A few hours ago, it was

In linear algebra, a projection is a linear transformation from a vector space onto a subspace which does not move the points in the subspace.

Both definitions are correct, but I prefer the second formulation, because it matches better with my intuitive understanding of the term projection. Comments? -- Jitse Niesen (talk) 09:03, 25 October 2006 (UTC)

yeah, in the intro i'd prefer something more intuitive, too. i took out the second one, though, because it's incomplete and inaccurate. i couldn't come up with something better at the moment.  :( maybe i'll get a chance to poke through some textbooks to get an idea of how better to explain it (precisely and concisely).

and, btw, i'm not particularly attached to how things are worded or organized so feel free to muck with the article. Lunch 19:14, 25 October 2006 (UTC)

What is incomplete/inaccurate about the second definition? The "does not move" is perhaps not totally clear. but for the rest I don't see any problems. -- Jitse Niesen (talk) 02:12, 26 October 2006 (UTC)

I think the second is plain speak and clearer and has more information. The implications of P² = P are not obvious unless you are farmiliar with them, it's sort of a side fact--a way of identifying a projector. As far as "does not move" that might sound better as "affect" but I like it the way it is. Lunch, can you tell me what is "inaccurate" in the second example? Pdbailey 13:22, 26 October 2006 (UTC)

again, i agree a more intuitive introductory paragraph would be nice.

well, OK, perhaps i'll strike "inaccurate" and roll my complaints into "incomplete".

first, "a projection is a linear transformation from a vector space onto a subspace". i might change this to "a projection is a linear transformation from a vector space V onto a subspace of V", but that introduces a symbol. there's gotta be a way to get that point across without using a symbol and without being wordy.

second, "a projection does not move the points in the subspace." that's fine (other than needing some wordsmithing). but it leaves the question: what does a projection do with everything else?

i s'pose i'm spending more time talking about this than doing anything ... sorry. Lunch 22:50, 26 October 2006 (UTC)

Not sure why the first sentence can't cover less than everything, how's this:

In linear algebra, a projection is a linear transformation from a vector space onto a subspace of that vector space. projections do not move points within the subspace that is their range so that if P is a projector, applying it once is the same as applying it twice and Px = PPx or just

P = P².

I know thats more than one sentance, but we have no need not to write more than that. Pdbailey 01:07, 27 October 2006 (UTC)

I gave it a try. Feel welcome to shoot it down. -- Jitse Niesen (talk) 11:32, 29 October 2006 (UTC)

[edit] abstract example makes no sense

I just tried to cleanup the abstract example, but it is so all over the place that I just can't. I wonder what the value of the section is at all.Pdbailey 01:30, 30 October 2006 (UTC)

What does "k <= n column entries, each of length n" mean? Just that there are k columns and n rows? McKay 01:40, 30 October 2006 (UTC)

yes Pdbailey 05:47, 31 October 2006 (UTC)

So I fixed this point, and improved the typesetting, but I still don't understand it. Both "span" and "kernel" are spaces, not matrices. I vote to delete the section unless someone can clarify it. McKay 06:25, 31 October 2006 (UTC)

I think I understand this section. It's not really an example, but the general form for oblique projections, and it generalizes the formula mentioned at the end of "orthogonal projections". However, I agree that this section is not very clear. -- Jitse Niesen (talk) 08:20, 31 October 2006 (UTC)

[edit] Projection operator

Anybody thinks it is worth merging this article with Projection operator (that is, redirecting that one to here)? Oleg Alexandrov (talk) 16:18, 31 October 2006 (UTC)

I agree, so I did. -- Jitse Niesen (talk) 00:57, 1 November 2006 (UTC)

[edit] eigenvalue proof

A recent edit removed the proof that a projectors eigenvalues are all one or zero... All I can say is that I used to wonder why professors always just did proofs in math classes until I got to graduate school and i realized that if they didn't do the proof, I had no way of judging if their claims were correct. I trust math references that have proofs for their claims and reject those that don't. But maybe I'm just an odd-ball. Pdbailey 04:11, 1 November 2006 (UTC)

Well, this is an encyclopaedia, making me reluctant to include proofs. The usual way to achieve verifiability here is to add some references (which are sorely lacking at the moment). The fact that the eigenvalues are all zero or one also follows from the next sentence ("The eigenspace corresponding to the eigenvalue 0 is the null space U, and the eigenspace corresponding to 1 is the range V"), because U and V span the whole space.

On the other hand, it is a short and cute proof. If, after reading my comment and Wikipedia:Manual of Style (mathematics)#Proofs still think that it belongs in the article, feel free to add it back in. -- Jitse Niesen (talk) 11:21, 1 November 2006 (UTC)

Thank you very much for pointing that out to me! You're right, it shouldn't be in there. Pdbailey 18:14, 5 November 2006 (UTC)

[edit] Properties Section

There is an error in the content here.The image has the caption

"The transformation T is the projection along k onto m. The range of T is k and the null space is m".

It should be the other way around, null space k and range m. Similarly, the last sentence reads

"Only 0 and 1 can be an eigenvalue of a projection. The eigenspace corresponding to the eigenvalue 0 is the null space "U", and the eigenspace corresponding to 1 is the range "V"

Throughout the section he refers to the properties of the nullspace V, and range U.

I have corrected these problems, they read correctly now.

Thanks. If someone gets the time, the notation in the image should be made consistent with the notation in the text (which could probably stand some reworking itself). Lunch 21:57, 13 November 2006 (UTC)

[edit] Non-linear subspaces

In my work, I use the terms "projection" and "projector" for these mathematical objects which project onto any set, and not only on linear or affine subspaces. I am sure there is some formal mathematical literature on that, although I only know references of this usage in more applied contexts. Shouldn't the definition be wider (that is, some non-linear transformations could also be called projections)? In metric spaces, one can further define a "distance-minimizing" projection, which is not only idempotent but also maps any point of a space to the point in a set that is closest to it (this can be seen as an extension of the "orthogonal" projection, I guess). Anyway, I just wanted to raise this point of view, I don't know to what extent this extended definition of projection is wide-spread enough to include it. Pierre T. 13:36, 19 March 2007 (UTC)

seem like fair enough comments. if you, or some other knowledgable folks, got some references and are convinced that it is sufficiently common usage, i'd suggest add it to the article (or possible create a new article). Mct mht 13:59, 19 March 2007 (UTC)

yup, there are other notions of projections, and if you add material about them, that'd be great. this particular article is about the use of projections in linear algebra, though. it'd probably be best to start another article; the addition would be very welcome. cheers, Lunch 01:50, 20 March 2007 (UTC)

Ok. I'll look into that. I now came to realize that there are no wiki entry on the convex feasibility problem... There is quite a large body of work on this, and people often use projections onto convex sets in the formulation of their algorithms. That might be the right place to start. Thanks for your encouragements.Pierre T. 13:35, 20 March 2007 (UTC)

[edit] Orthogonal Projection Section

I think 'if x is a vector in the domain X of the projection', should say 'image/range of projection'; X is not the domain as described. Also (P-P^2)=0 is not sufficient for orthogonal projection I don't think, just projection. I think, transpose(P)*(1-P) should do.

seems like it should just be domain, the "X" shouldn't be there. for your second point, article could be more clear there as well. the assumption throughout is, apparently, that P is self-adjoint, which is correct (notice self-adjointness is invoked in the first equation of the section). but perhaps that needs to be worded more explicitly. Mct mht 22:31, 25 March 2007 (UTC)

what that paragraph is saying is that a Hermitian matrix P is a projection (P^2 = P) if and only if it is an orthogonal projection. that doesn't look very helpful. a more appropriate point to make would be "a projection is orthogonal if and only if it is Hermitian." Mct mht 22:45, 25 March 2007 (UTC)

That is the point which I intended to make, though I wrote it in a terrible way. Thanks for fixing it. -- Jitse Niesen (talk) 12:07, 27 March 2007 (UTC)