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It was wrong to redirect eigenvalue to eigenvector. Vectors aren't the only things that can have the, as it were, "eigen-" idea applied to them. For instance, eigenfunctions have eigenvalues. PML.

I agree that it is wrong to redirect eigenvalue to eigenvector and I have corrected this. But your reason is incorrect. Eigenfunctions are vectors in the linear space of functions. wshun
I cannot but quote :o) salvatore.federico

Vectors and values are closely related. The complete basis, or superposition principle is ubiquitous in linear algebra. Each vector (or function in the broadest sense) of a normal complete set has an associated value. The best practical applications have +ve definite values. These could be electron energy levels, or frequencies of a vibrating plate. — Preceding unsigned comment added by (talk) 06:12, 3 December 2014 (UTC)


Note that for infinite dimensional Hilbert spaces, eigenstates only exist in an extension of that space. For example, Dirac delta functions only exist in an extension of L2(R).

Is this really true for all eigenvectors of all linear operators? What about, for example, the stationary states of the hydrogen atom, which are eigenvectors of the Hamiltonian? This is not fare to edit wikipedia because it creates data inconsistency and people usually don't take it as authentic resource of information. — Preceding unsigned comment added by (talk) 13:42, 18 May 2016 (UTC) Josh Cherry 21:35, 14 Oct 2003 (UTC)

No, this is nonsense --MarSch 1 July 2005 20:55 (UTC)

Origin of the term "eigen"[edit]

Eigen has some different meanings in German, and there's also a nobel prize winner with that name. In the FAQ of the German newsgroup de.etc.sprache.deutsch [1] they state that the names for "eigenvalue" and "eigenvector" come from the meaning "inherent, characteristic". nikai

Can anyone confirm that the German (or Dutch) word "eigen" is indeed the correct etymology of the words eigenvalue and eigenvector (besides the German etymology faq from above)? It would suggest that a German or Dutch mathematician was the first to use these terms, but I can't find any references of this. Who was first to name an eigenvector an eigenvector? --Anthony Liekens 15:04, 1 Jan 2005 (UTC)
According to [2], David Hilbert first used Eigenwert in 1904. Hermann von Helmholtz coined similar words earlier (like Eigentöne, which were translated as proper tones). -Nikai 14:42, 17 Mar 2005 (UTC)
Eigen means own, as in "my own vector". --MarSch 1 July 2005 20:57 (UTC)

I coppied the following text from a dictionary defining peculiar [The Collaborative International Dictionary of English v.0.48].

One's own; belonging solely or especially to individual; not possessed by others; of private, personal, or characteristic possession and use; not owned in common or in participation. [1913 Webster]

I think this would also be a good definition of eigen, so the best translation of eigen might be peculiar.

"Das ist mein eigenes" - "That is my own"


"Er ist eigen" - "He is peculiar"
"Eigenheit" - "pecularity"

I would not translate it as proper, but maybe I have a wrong comprehension of proper.

It is very unlikely, that it is named after the nobel prize winner.

I am a native German speaker. The Dutch meaning of eigen probably differs slightly from the German meaning. Markus Schmaus 4 July 2005 01:15 (UTC)

I think 'proper' is the most correct word to translate 'eigen' in this case, 'peculiar' also means 'strange' which is not appropriate, 'own' is used for persons, not for vectors. I'm a native dutch speaker and I know 'eigen' can be translated perfectly by 'proper'. What do native english speakers think about the use of the word 'proper'? Michigun (talk) 00:45, 18 July 2008 (UTC)

I am a native Dutch speaker. In Dutch, eigen means own, private, innate, natural. Although it does not mean peculiar, the Dutch have a closely related word eigenaardig, which does mean peculiar. [[User:Loe Feijs]|]] 12 August 2005.

I agree that peculiar is the best translation. For reference, Hilbert wrote:
"Insbesondere in dieser ersten Mitteilung gelange ich zu Formeln, die die Entwickelung einer willkürlichen Funktion nach gewissen ausgezeichneten Funktionen, die ich Eigenfunktionen nenne, liefern: ..."
David Hilbert, Grundzüge einer allgeminen Theorie der linaren Integralrechnungen (Erste Mitteilung), Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, 1904, pp. 49ff (quote on top of p. 51)
The word ausgezeichnet (excellent) seems to support the peculiar translation.
In Dutch, eigen only means own and not peculiar, but that is not relevant. See also the eigen lemma on LEO. -- Jitse Niesen (talk) 4 July 2005 12:43 (UTC)
I don't believe ausgezeichnet means excellent in this context. It means particular as far as I can see. I don't see any clues to the meaning in this context. On the other hand I understand why he called them eigenfunctions, because they are somehow intrinsic/geometric. --MarSch 4 July 2005 13:40 (UTC)
I think in this context ausgezeichnet means something like particular, special or, maybe the best translation, distinguished. Markus Schmaus 4 July 2005 14:04 (UTC)

Most translators will tell you that eigen in German translates to characteristic in english. This makes sense since and eigenvector is sometimes refered to as a characteristic vector. So whats all the confusion about? --Jcrocker

Do you have any references to support your first statement? It is true that eigenvector is sometimes refered to as a characteristic vector, but that does not mean that characteristic is the correct translation of eigen. In fact, characteristic is derived from the works by Cauchy in French. -- Jitse Niesen (talk) 09:13, 13 July 2005 (UTC)
Could you give a German sentence in which eigen is best translated as characteristic? Markus Schmaus 14:54, 13 July 2005 (UTC)
10 years late to the party, but just browsing up on the etymology of Eigenwhatevers I stumbled across this discussion. A possible sentence (albeit a not-really-common one) could go along the lines of "Diese Technik ist Goethe eigen." probably best translated as "This is a characteristic technique of Goethe". Of course it directly derives already from the translation "own" mentioned above. And generally translating "eigen" to "characteristic" would be stretching it. --Ulkomaalainen (talk) 01:28, 8 April 2015 (UTC)

I think the term "Peculiar" may be the most apt, as this seems to pertain both to eigenvectors and the "characteristic" polynomials that are used to derive them. Eigen as applied to vector, value, function seems to possibly mean "particular" or "peculiar" or "characteristic" to, as applicable, a matrix, lambda value, or set. --frogstomper

I agree that "peculiar" makes the most sense, and in fact it used to mean exactly what eigen seems to mean, see the Etymology Online definition. The "unusual" meaning seems to be fairly recent. -- meganite —Preceding unsigned comment added by (talk) 17:08, 20 October 2009 (UTC)

What are they good for?[edit]

How about adding some examples what eigenvectors can be used for? Some simple examples for a non-mathematician to understand why we need to know about them. Are they useful for programming, for example? 3D graphics?

yeah good point since the only thing I know is within solving DEs. (talk) 08:56, 10 May 2012 (UTC)

I use term to refer to the "eigenvector" of an incoming request from a host in TCP. When I want to connect to a host, I will use my eigenvector, which I think of as "my unique key" or "my idiosyncratic vector" which is able to traverse the boundary because it is aimed correctly at the target. More specifically, in idiosyncratic vocabularies, two might join at one idiosyncratic word in both wordballs. That is, the wordballs are connected through an eigenvector which properly triangulates on another wordball via the eigenvector "menschenkenner" or "Weltanschauung." If I use "Weltanschauung" [and "Umwelt"] and "Naamitapiikoan" I would establish an eigenvector tying together "Maslow" and "Siksika." As you see it, is that proper? Please comment or delete at will. — Preceding unsigned comment added by (talk) 20:54, 13 October 2013 (UTC)

Here is a good source [3] which explains why Eigenvalues/Eigenvectors are useful. They are essentially the steady state of feedback loops, or Morkov Chains, that can be represented using linear transformations. — Preceding unsigned comment added by 2601:681:4F01:8BC0:EC52:2C37:8599:2092 (talk) 23:15, 30 May 2018 (UTC)

Not everything is decomposable[edit]

It would be nice to have some link to singular value decomposition via discussion of eigenvectors of : linas 1 July 2005 14:29 (UTC)

merge with eigenvalue[edit]

Apparently this has already been discussed in the past, but I cannot help but notice that eigenvalue has a lot of material which duplicates this articles (or vice versa), unsurprisingly. The one cannot be described in isolation of the other, thus they should be one article. Also eigenspaces should be in this article. Suggested name for this article eigenvalue, eigenvector and eigenspace. --MarSch 1 July 2005 21:37 (UTC)

I am not in principle opposed to the move. I would like to note only that the articles are in rather good shape now, and besides, they also have a lot of nonoverlapping material, and rather different emphasis.
Therefore, please note that merging them will be a lot of work, and since the articles look good the way they are now, nothing less than highest quality of the finalized article will make me happy. If you feel you can devote several days of your life, for combined 8-10 hours to do this, be my guest. Otherwise, there are plenty of other things on Wikipedia needing work than merging these two articles. Oleg Alexandrov 2 July 2005 05:34 (UTC)
Are you kidding? These articles are in pretty bad shape. --MarSch 2 July 2005 14:24 (UTC)
Then how about putting a merger tag? so that others who are willing (i.e., have time) can work on the merger. -- Taku July 2, 2005 07:05 (UTC)
I will put the merge templates up. --MarSch 2 July 2005 14:24 (UTC)
I worked a bit on the articles some time ago, and actually I also considered merging them. My main reasons for not doing so were that it was apparently discussed before, that I was afraid the merged article would become too big, and that I was lazy. But I think the articles could well improve after a merger (though I would not mention eigenspaces in the title). In response to Oleg I'd like to state that these articles are indeed better than average, but they are also far more important than (for instance) inverse quadratic interpolation, so any time spent improving them, however slightly, is well-spent. -- Jitse Niesen (talk) 3 July 2005 02:15 (UTC)
Eigenspaces are arguably more important than eigenvectors. Especially when they are not 1-dimensional they are more intrinsic than (a base of) eigenvectors. --MarSch 3 July 2005 12:00 (UTC)
I would like us to first discuss the newly created eigenvalue, eigenvector and eigenspace article before redirecting eigenvalue and eigenvector to it. Just to make sure we agree on what is there. Oleg Alexandrov 3 July 2005 15:51 (UTC)
Hi Oleg, thanks for rewording your reply ;) I wasn't planning on redirecting until the new article is finished. This way of working manifold suits me very well. I know there is still a little argument about the title, but I figured we can always move it once we decide what it is and until then the new title is sufficiently big that no accidental wikilink will ensue. Unless you think that is not enough. I hope this puts your mind at ease. --MarSch 3 July 2005 16:25 (UTC)