Where does the nomenclature "eigen" originate?

[kostoglotov]

In eigenvectors, eigenvalues, eigenanalysis...why the word eigen and what's the history, how did these things come to have the prefix eigen?

My best hunch is that the German word "eigen" means "own"...but that's not very illuminating.

 

[fresh_42]

In eigenvectors, eigenvalues, eigenanalysis...why the word eigen and what's the history, how did these things come to have the prefix eigen?

My best hunch is that the German word "eigen" means "own"...but that's not very illuminating.

It's German, yes. And as often you can't match words of different languages 1:1. Eigen is usually a part of words and seldom used on its own. On its own it means something like being special. But as part of other words like Eigenwert (eigenvalue), Eigenvektor (eigenvector) or Eigenraum (eigenspace) it means "belonging to" as a property. It's simply a 1:1 emigration. In German it makes a lot of sense. As the zeros (Nullstellen, "zero points") of the characteristic polynomial eigenvalues do really "belong to" a linear function. Plus it doesn't seem to make a problem for English speakers. So they probably didn't want to create something totally new.

 

[jtbell]

My best hunch is that the German word "eigen" means "own"

Correct, that's the basic meaning. In mathematical usage it might be translated into English more literally as "characteristic": Eigenwert (Eigenvalue) = characteristic value, etc. But two syllables are faster to say than five.

 

[fresh_42]

Btw there are a couple of more words in German having "Eigen" as prefix.
Eigentum means property in the sense of ownership
Eigenschaft means property in the sense of attribute
Eigenart means the nature of things
eigenartig means strange
Eigenanteil means the share that one personally carries, e.g. in insurance contracts
Eigenkapital means equity capital
... and some more.

So in German it's kind of natural to talk about eigenvalues. I remember my surprise when I first heard it in English.
I thought someone used it and didn't know the correct English word.

 

[fresh_42]

Wikipedia states that the term is originated in an article by David Hilbert, 1904:
"Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen. (Erste Mitteilung)"
(Fundamentals of a general theory of linear integral equations. (First report))

"At the start of the 20th century, Hilbert studied the eigenvalues of integral operators by viewing the operators as infinite matrices. He was the first to use the German word eigen, which means "own", to denote eigenvalues and eigenvectors in 1904, though he may have been following a related usage by Helmholtz. For some time, the standard term in English was "proper value", but the more distinctive term "eigenvalue" is standard today.

J. Dieudonné mentioned in his "Abrégé d'histoire des mathématiques 1700-1900" that first examinations of determinants and eigenvalues, although not named as such, date back to Lagrange's work on partial differential equations (1762) and J.C.F. Sturm - J. Liouvilles work on operators in the first half of the 19th century..

 

[jack476]

It comes from German, it's an adjective prefix that refers to something characteristic or particular, as in a unique characteristic trait of a person, object, etc. Hence why you see eigenvalues sometimes called "characteristic values" or why the way to find eigenvalues for really simple (homogeneous linear constant coefficient) 2nd order ODEs is called the method of "characteristic polynomials".

https://en.wiktionary.org/wiki/eigen-

 

[f95toli]

In some languages there is a direct translation of "eigen" and then that is used. In Swedish eigenvalue is translated as "egenvärde"; "egen" is the Swedish version of "eigen" (and is also used other contexts German "eigenart" translates as "egenart", "eigenschaft" as "egenskap" etc).
It it is a bit strange that there is no good English word for this since there are English versions of most Germanic words.

 

[fresh_42]

Own and eigen share the same etymological origin.
http://www.etymonline.com/index.php?allowed_in_frame=0&search=own&searchmode=none

I think that the real advantage in languages as Swedish or German here is, that you can build composites like egenskap or Eigenwert whereas ownvalue would have been to be written in two words: own value. As mentioned earlier they have tried with the more English like construction: proper value. Since eigenvalue is in deed more distinctive it became the standard.

 

[fresh_42]

Own and eigen share the same etymological origin.

The difference between the two is their "Eigenleben".
However, they have developed differently ever since. While the German "eigen" kept its broad meaning that describes property and ownership on one hand as well as attribute and specialty on the other, the English "own" drifted towards property in its meaning of ownership alone.
I guess that's the true reason behind this immigration. It would be delightful to debate what this fact (change of meaning) says about English speakers.

Btw.: Yesterday I've read in an article the word "Ansatz". That was really weird.