Talk:Projection (linear algebra)

Fix Complementary spaces in oblique projection section please
The oblique projection section repeatedly calls the range and null space complementary spaces, when of course the 1) range and left null space, and 2) row space and null space are the correct complementary spaces. Can somebody qualified make the changes?

Projection?
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

"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)

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)

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 $$P^2=P$$ 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 $$Px = P P x$$ or just

$$P = P^2$$.


 * 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)

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)

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)

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 Manual of Style (mathematics) 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)

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)

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)

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)

Projection operator properties
I didn't found the following pretty useful property of a projection. I'd like to know if people would like to add this property on the page and if there would be any needs for a proof.

Property: If $$P:V \longrightarrow V$$ is a projection than $$tr(P) = dim(Im(P))$$

--Noud 18:46, 30 November 2007 (UTC)

i suggest we leave it out. in the finite dim case, that's a trivial fact due to unitary invariance of trace. in the infinite dim case, a projection need not be trace class and one needs to be more careful. Mct mht 22:22, 30 November 2007 (UTC)

Start with simpler examples
Start with vector-vector projections as people learn in physics and early on in linear algebra. This article is too technical for people looking for such basics, they serve to help build intuitions, and this would be better form for a general encyclopedia. The transformation view, while correct, is the wrong way to start the article. —Preceding unsigned comment added by 71.111.195.53 (talk) 03:27, 30 April 2008 (UTC)

Facts vs. explanations
This article is nearly worthless. It is a good example that stating a lot of facts is *not* the same as explaining something.

-- Bruce Jerrick —Preceding unsigned comment added by 76.27.194.188 (talk) 00:22, 7 May 2008 (UTC)

I totally disagree — Preceding unsigned comment added by Antoniojpan (talk • contribs) 11:48, 2 February 2020 (UTC)

Uniqueness
Is there an error in "...decomposition ..into direct sums is not unique in general. Therefore, given a subspace V, in general there are many projections whose range (or kernel) is V." yet earlier, it says the subspace U and its complement V determines the projection P uniquely which I think is correct. 218.186.13.2 (talk) 16:52, 3 September 2008 (UTC) Rasjid


 * Given both the range U and the kernel V, the projection is unique. However, if only V is given then the projection is not unique because there are many possibilities for U.
 * I think the text of the article is correct, but perhaps you can make it clearer? -- Jitse Niesen (talk) 04:58, 4 September 2008 (UTC)

rewrite of Classification
deleted LearningQM (talk) 13:55, 5 September 2008 (UTC)

Classification
ignore - deleted LearningQM (talk) 13:53, 5 September 2008 (UTC)

Infobox
I have removed the infobox added by User:SharkD (diff). In my opinion the infobox is only marginally related to this article as it discusses projections from a graphical or artist's point of view, whereas this article is theoretical. On the other hand the infobox detracts from the article by forcing the first figure to move down so that it is not visible while looking at the lead. The figure is a good intuitive explanation of the concept and ought to be the first thing seen when this page is opened. --Zvika (talk) 05:30, 15 January 2009 (UTC)
 * I agree, I also don't like this box on top of all the other pages it is on - someone ought to convert it to a standard navigation box (at the bottom of the page) as is used on all other comparable pages instead. --Allefant (talk) 11:03, 15 January 2009 (UTC)

Could use better examples, with pictures
I think this article could be improved by adding a couple of examples in 2- and 3-d with pictures. The existing Simple Example section is a good start on this, but it lacks pictures, and could use some copy that takes more of a geometric or intuitive tone, than the math-term laden function which maps the point (x, y, z) in three-dimensional space $$R^3$$ to the point (x, y, 0) is a projection onto the x-y plane. Imagine being a newbie, and reading things like function which maps. Personally, as a college freshman (in engineering!), I would have been a little bit confused about that, let alone a layperson, who isn't familiar with the concept of a function as a map from space to another. I will make this change some day, but that day will likely never come, to be frank. User:!jimtalk contribs 06:58, 20 October 2009 (UTC)

a little error
There is a little error. The inverse (lambda I - P)^{-1} it is 1/lambda I + 1/lambda * 1/(lambda-1) P

Proof: If we call L=lambda: (L I - P )^{-1} = 1/L (I-P/L)^{-1} = 1/L sum (P/L)^n = 1/L *I + 1/L *P/L + 1/L * P/L^2 + ... = 1/L I + 1/L (1/L + 1/L^2+ ... ) P = 1/L * I + 1/L(L-1) * P

Cheers. — Preceding unsigned comment added by 95.120.230.142 (talk) 03:24, 8 January 2012 (UTC)

Mistake in Banach Space Section
I will delete the sentence "However, every continuous projection on a Banach space is an open mapping, by the open mapping theorem." This is simply wrong. Suppose that this is true. Then every continuous projection P maps the whole space X onto an open set. But PX is a closed subspace and can only be open if it coincides with X. Hence, PX = X which implies P = I. In other words, the only continuous projection is the identity (which is a contradiction whenever dim X is at least 2). — Preceding unsigned comment added by 92.225.117.200 (talk) 14:39, 26 April 2014 (UTC)

Formulas subsection
The example shows an orthogonal projection onto a line and how the component orthogonal to u disappears. Shouldn't it be:

$$ uu^\mathrm{T}x_\perp = u\cdot 0 $$ not $$ \vec{0} $$? 130.75.31.239 (talk) 19:40, 19 June 2015 (UTC)

I think it should say: if we fix an orthogonal basis...

--antpan (talk) 11:49, 2 February 2020 (UTC)

Usually, an orthogonal projection is defined as self-adjoint projection (see Rudin's Book on Functional Analysis or Sakai's book on C*-algebras. The reason is that one wants the image space PH to be orthogonal _|_ to the kernel (1-P) H. in that case one needs a self-adjoint Matrix/Operator P. Then one has:


 * $$=<(1-P)Px,y> =0$$

LMSchmitt 22:59, 18 December 2020 (UTC)

Orthogonal projection. Proof of existence
It is not clear what the "Proof of existence" is actually proving, because no statement is given. Is this a proof of the fact that an operator mapping a point in a Hilbert space to the nearest point within a closed linear subspace is an orthogonal projection as previously defined (e.g. linear, idempotent and self-adjoint)? If so, this statement must be formulated somewhere. Preferably, right before the proof. AVM2019 (talk) 18:15, 17 December 2020 (UTC)


 * shown: every closed subspace S in a Hilbert space H generates an orthogonal projection P onto S such that the orthogonal space S' in H corresponds to (1-P)=P'

LMSchmitt 18:42, 20 December 2020 (UTC)

Error in orthogonal projection existence proof
The current proof of existence of orthogonal projection claims

"For every $$\mathbf x$$ the following set of non-negative norm-values $$\{\|\mathbf x - \mathbf u\| \mid \mathbf u \in U\}$$ has an infimum, and due to the completeness of $$U$$ it is a minimum."

Which confuses completeness and compactness; this is not sufficient to establish the existence of a minimum (or uniqueness). For an example of a Lipschitz-continuous function from a complete bounded metric space to $$\mathbb R$$ which does not attain its infimum, consider $$d(-,A) : \bar B_{\ell^\infty}(0,1) \to \mathbb R$$ with $$A = \{(1+2^{-i})e_i \mid i \in \mathbb N\}$$ for $$e_i$$ the standard Schauder basis. (I do have a proof, but that uses approximations by projections to finite dimensional subspaces, and if there's a fix that doesn't require entirely changing the structure of the proof that would work better.) Emlili (talk) 09:57, 27 April 2021 (UTC)

Just realised that Hilbert projection theorem has a working proof (at a glance) so I'll link it in the article, but I've not changed the proof here. Emlili (talk) 10:24, 27 April 2021 (UTC)


 * WRONG. a) there is no use of compactness here. This works also in oo-dim HSs. b) The proof is as in Hilbert projection theorem as you found out. LMSchmitt 11:36, 28 April 2021 (UTC)

Complete inner-product spaces
The article mentions that if we have a complete inner-product space $$V$$, then we can say a projection $$P\colon V\to V$$ is orthogonal to mean $$\forall x,y\in V\langle Px,y\rangle=\langle x,Py\rangle.$$ However, this definition does not require completeness. Does anything go wrong if we remove from the article the assumption of completeness? Thatsme314 (talk) 14:06, 17 October 2023 (UTC)


 * I'd assume that typical sources only assume completeness to talk about adjoints, and they might define orthogonality as self-adjointness of $$P$$. Thatsme314 (talk) 14:09, 17 October 2023 (UTC)