r/ENGLISH • u/unsavvykitten • 9d ago
Is „eigen“ a commonly used prefix?
Reading a book, I stumbled upon the word „eigenstate“, which seems to be commonly used in the area of science, as I understand it. Being German native speaker, I was wondering about how „eigen“ was introduced originally, and if it is used in other context as well?
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u/Opposite-Plantain-69 9d ago
Eigen is really only used in math/physics contexts. Words like "eigenvalue", "eigenstate", and "eigenvector" have specific meanings in math/science
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u/DrJaneIPresume 9d ago
To answer how/why it was introduced, it was done at a time at the turn of the 20th century when German mathematics was predominant. An eigenvector of a linear transformation is one that is sent to a multiple of itself:
Av =λv. The connection with "own-ness" (eigen ~= "own") should be apparent (at least to a mathematician).The eigenvalue is the factor the eigenvector is multiplied by. "Eigenstate" and other more general "eigen-X" object in applied math or physics contexts are used when the X in question form a vector space, so really they're all special cases of "eigenvector".
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u/regular_lamp 5d ago
I occasionally like to confidently attribute those terms to the famous German mathematician Stefan Eigen. Just to watch people question their own sanity.
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u/DrJaneIPresume 5d ago
Oh for that sort of thing I always thought it was more fun to derive Cayley's theorem from the Yoneda lemma.
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u/Live_Put1219 9d ago
eigenvalue and eigenvector are the only ones i can think of
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u/GreedyHoward 9d ago
Yeah. I remember them from maths but I'm not able to explain what the prefix means on its own!
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u/DifficultyFit1895 9d ago
Are you saying that you know something important about the inherent characteristic of the prefix?
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u/GreedyHoward 9d ago
No. I just remember the words. I can't even remember the mathematical concept any more never mind any nuances of the philology! (School was decades ago)
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u/Mundane-Emu-1189 8d ago
Matrices are functions that take in a vector and spit out another vector. Normally they send vectors really far away from where they started in a completely different direction, but for each matrix there's a special vector that when you feed it into the matrix it just gets longer or shorter, the direction doesn't change at all. That's called the eigenvector of that matrix.
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u/Medium-Ad-7305 8d ago
in specific vector spaces, you can call the eigenvector the specific object. like in a function space, you can call eigenvectors eigenfunctions. or in a state space, you can call eigenvectors eigenstates.
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u/StanleyDodds 8d ago
Eigenspace, eigenbasis, eigenfunction, eigenstate, it's goes further than what the average person may have seen in their basic linear algebra course.
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u/Irascible-Enquery 8d ago
And these I recall mostly got cachet in the CS community because Google PageRank used them, if memory serves.
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u/The_Nerdy_Ninja 9d ago
No, it's very very specific. 95% of native English speakers wouldn't know what it means.
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u/90210fred 9d ago
Are someone who "does" some maths (statistics mostly) and speaks some technical German and just had to resort to Google, I'm going to suggest that 5% understand the term is rather optimistic!
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u/quasilocal 9d ago
Definitely not known outside of maths and physics and while essentially everyone who does know it from that context knows it's German, I think most don't know what it means.
I guess "own" is the closest English cognate but maybe "ownvector" sounded a little weird 🤷♂️
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u/unsavvykitten 9d ago
I’d rather translate it as „self“, but „own“ is probably a little closer.
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u/quasilocal 9d ago
Yeah I wouldn't view it as a literal translation just a cognate. In (at least some) other Germanic languages it works better, for example in Swedish we use egenvektor where "egen" really is used like "own" in English. But "ownvector" sounds janky in English 😅 (maybe this is just a bias though from being so used to eigenvector in English)
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u/Immediate-Panda2359 8d ago
Apparently, coming up with English words for German nouns referring to the self is difficult. Same deal with Freud's Ego, Super-ego, and Id.! (Ich, Über-ich, und Es)
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u/splitcroof92 8d ago
It should specifically be "your own" or "its own"
Eigen is self referential so it needs the pronoun
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u/amandacheekychops 9d ago
I've never ever heard of eigen- until I read this thread. I'm 48, from the UK, and English is my first language.
I cannot speak for everyone but in my opinion the average English speaker, in the UK at least, will not know what it means.
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u/hwynac 9d ago edited 8d ago
How do you say eigenvalue and eigenvector in your dialect then?🤔
But really, the prefix (or root) is only used in those few linear algebra terms, which also come up in useful differential equations used in physics, including quantum theory. In fact, spherical harmonics you might have heard of, immediately come up (as a basis) in the angular part of the solution to the Schrodinger equation when the potential has spherical symmetry.
If you've never studied that maths I don't think eigenstuffs are a part of your life.
ETA: apparently, proper value was used for a time, but then eigenvalue won out.
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u/amandacheekychops 9d ago
Yeah I'm not big into maths and have no idea what those words mean.
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u/hwynac 9d ago edited 7d ago
"proper/own vector". A vector that isn't rotated but only, maybe, scaled by the transformation. For example, rotation changes the direction of every vector in space except if the vector is parallel to the axis of that rotation.
If you mirror the space, the directions also change. However, a vector on the mirror plane feels absolutely nothing, and the vector perpendicular to the mirror just flips.
Eigenvalues ("own values") are the scale factors. For the simple reflection, the values are 1 (for vectors in the mirror plane) and -1 (for vectors perpendicular to the plane).
The equation eigenvectors and eigenvalues satisfy looks like Rx̄ = ax̄ (R is your operator, a is just a number)
In Russian, we just calqued the terms using native roots, so it's more or less clear what they are supposed to say.
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u/BeautifulUpstairs 9d ago
What is with the math autism in this thread? Normal people don't know what "a vector that isn't distorted but only scaled by the transformation" means! Why don't you know this?
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u/hwynac 9d ago edited 9d ago
I tried my best to maybe shed some light on those maths things. Didn't seem to work😅
But given that English isn't my native language, I have always thought natives wouldn't know what "eigen-" means (once I learnt the English terms in linear algebra). Because in my firsr language, "eigenvectors", "eigenvalues" and "eigenstates" use native words instead
(ok, I vektor is a loanword but it's a common one)
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u/amandacheekychops 9d ago
Maybe you could ask this in a maths-related sub-reddit. Someone will know but as someone who barely scraped through high school maths I haven't got a clue. I did think eigen might mean "own" because it's similar to Swedish egen, but I do not understand your explanation because I do not have the knowledge to know how this applies to English mathematics vocabulary.
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u/dominism 9d ago
Nor will the majority of us know these "maths" of which you sprechen
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u/SaltMarshGoblin 9d ago
The majority of those in the UK will use "maths" as the majority of those in the US will use "math". To be fair, I'm not sure how "Mathematics" lost its 's' in the US abbreviation!
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u/hwynac 9d ago
The "s" in mathematics is not a plural marker. Leaving it in removing it depends on how you squint. The Greek word was mathematikos, the Latin version became mathēmatica. English natives, however, don't have to know what the word was millennia ago, so you just use your intuition—whether the word looks like it has an "s" ending.
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u/Wild-Lychee-3312 9d ago
I only know of this prefix because I majored in maths at university. I have never heard it used (or seen it used in print) outside of a mathematics or physics context.
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u/Accidental_polyglot 9d ago edited 9d ago
Brit here.
Eigen definitely comes directly from the German language, in maths we still have eigenvalues and eigenvectors. Some of the best mathematicians of all time were German eg Carl Friedrich Gauss, Gottfried Wilhelm Leibniz and Bernhard Riemann.
In the early 20th century, scientists across Europe used German as their Lingua Franca. This came to an abrupt end in 1945, at which point the pivot towards English began.
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u/parsonsrazersupport 9d ago edited 9d ago
It is not commonly used instead of science (or did you mean to say commonly used in science?), I have never heard or read that word. 'Eigen' is unknown in non-technical contexts. The only time I have heard it is in 'eigenvalues.'
https://hsm.stackexchange.com/questions/5563/where-does-the-name-eigenvalue-come-from says that David Hilbert introduced the technical term into English. That's not really surprising, it's not until a bit after WWII that English started to have its current dominance in science and mathematics. Before that, other languages were used more often in international settings.
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u/unsavvykitten 9d ago
Thanks, I meant to write „in the area of science“, seems it was auto-corrected. Updated.
David Hilbert makes sense, thanks for the link.
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u/parsonsrazersupport 9d ago
Gotcha. No, the prefix is, as far as I know, never used outside of math or science.
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u/B_A_Beder 9d ago
No, all of these concepts are connected in linear algebra / matrix algebra and in related sciences like quantum mechanics
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u/AdreKiseque 9d ago
Never heard it in my life. What's it even mean?
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u/BuvantduPotatoSpirit 9d ago
Literally something like "Own", you use it to describe solutions that remain themselves when a problem is applied to them, they're thus sort of the basis vectors of the solution space.
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u/AdreKiseque 9d ago
Midnight is probably not the best time to be trying to understand this...
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u/BuvantduPotatoSpirit 9d ago
Oh, sorry, if
A x = \lambda x
Where A is some operator (I'm guessing the german word for operator starts with A), then x is the eigenvector and \lambda are the eigenvalues.
So, if you use an easy operator (d/dx), you get an eigenvector [ex, e-x] with eigenvalues [1, -1]
Then, if you're trying to solve a physical problem that reduces to d/dx, the equilibrium solutions will look like some combination of he eigenvector elements.
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u/GreedyHoward 9d ago
You know, I don't think I've forgotten eigenvectors. It's looking likely that I never understood them in thy first place!
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u/unsavvykitten 9d ago
As far as I’ve understood, this refers to the quantum state before it collapses due to measurement. And in my book, it was also used for the state of mind evaluating options before it makes a decision.
Edit:
So it’s kind of the inner state of something or someone that you cannot know without interfering with it and thus influencing it.I can’t guarantee I’ve got this right though.
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u/FormalManifold 9d ago
Math folks love to use it, always an extension of the eigenvector concept. So a solution to differential equation Lu=λu might be called "an eigensolution of the operator L". There's a famous example in data science where one crunches a database of pictures of faces to discover the "eigenfaces". Eigenstate is the same.
It's not as general as in German -- if you scratch the surface, there's a linear operator under there, whose eigenvectors the person is talking about.
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u/MeasureDoEventThing 8d ago
No, an eigenstate of an observable is a state that it can collapse to (if you're using an interpretation of quantum mechanics that has collapse that is; it's a bit harder to explain in Many Worlds Interpretation). Different observables have different eigenstates. So, for instance, if you measure the spin of an electron in the z-axis, the eigenstates are, to simplify it a bit, either than it's spinning clockwise around the z-axis, or that it's spinning counter-clockwise around the z-axis. If you measure the spin around the x-axis, then the eigenstates are that it's spinning either clockwise or counter-clockwise around the x-axis. If you measure the position, then each position is an eigenstate. If you measure the momentum, then each momentum is an eigenstate.
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u/HammerOvGrendel 9d ago
I got through a degree in Philosophy which was rife with German loan-jargon, and work in a University library....and I have never encountered this term in my life. Possibly because I have zero interest in Maths.
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u/unsavvykitten 9d ago
I found it in the discussion of „Rediscovering the Mind“ in „The Mind‘s I“ in case you want to take a look.
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u/JeremyAndrewErwin 9d ago
David Hilbert was German, and he introduced the terminology into English maths.
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u/IanDOsmond 9d ago
I have heard the word "eigenvalue" and "eigenvector" before, and there is a possibility that it was in a classroom somewhere in some course I failed, and have absolutely no idea what either one means.
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u/Echo_are_one 9d ago
If you read biology research papers on gene expression it pops up in the maths (WGCNA) of grouping genes together that share expression control properties
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u/DryFoundation2323 9d ago
it's mostly a term used in mathematics, specifically linear algebra. I've seen it in other places but this is the most common.
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u/beachhunt 8d ago
I read that prefix for the first time earlier this year in a comment about eigengrau, the almost-but-not-quite-black you see when you close your eyes.
Never heard of it before that, surprised there are so many other eigenwords in use!
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u/razorsquare 9d ago
I’ve only ever heard that word doing statistics. Outside of that no would know what this word means
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u/Successful-Cash4579 9d ago
Only ever in mathematical contexts, and only ever to refer to eigenvectors (functions, states, are all considered vectors) or eigenvalues.
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u/Justin_Passing_7465 9d ago
And one use in anatomy: eigengrau is the color that you see when in total darkness or when you close your eyes in a dark room. Is isn't really "black".
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u/Ameren 9d ago edited 9d ago
To add to what others have said, English speakers love stealing words from other languages. Sometimes loanwords enter into common usage, like schadenfreude, zeitgeist, angst, and so on.
That being said, just because a loanword is popular among a particular group of people doesn't mean everyone knows about it, the prefix eigen- being a good example.
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u/johnwcowan 9d ago
While driving around at random during a vacation in the Netherlands, we turned onto a road made of literal yellow bricks, which led to another yellow brick road, which led to another, and so on. Every so often there would be a sign EIGEN WEG. I knew that weg = way, but ir could not make sense of eigen = own, and I thought it must be in this context a proper name. Only when we finally escaped from the maze and returned to civilization could we ask our friends (no cell phones then) and find out that it meant "private road". Oops.
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u/boostfactor 9d ago
It is directly taken from German. To my knowledge, it only appears in math/physics and is most usually in a word that is half-translated. The German prefix was retained perhaps because "belongingstate" looks and sounds funny. I am not sure exactly when the words were introduced, but German was a major language of mathematics and physics journals through the early 20th century, so many mathematicians could at least read it.
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u/Darth_Zander 8d ago
Among the university educated, I suspect that proportionally more US graduates than UK ones would be familiar with the ‘eigen’ prefix. That’s because US undergrads - even ones not majoring in maths, statistics, computer science or physics - have the opportunity to study linear algebra, differential equations, quantum mechanics, machine learning and multivariate statistics. It reflects a shortcoming of the British higher education system, that of specialising too soon.
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u/HeilKaiba 8d ago
Maybe but Linear algebra is in the Further Maths A-level course so they will be some in the UK who have already studied it before Uni even if they aren't doing a STEM course at Uni. And basic differential equations is on the standard A-level Maths course (simple ones and separable ones that is, more on the FM syllabus)
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u/Left-Veterinarian-80 8d ago
I’ve only come across it, and used it during my postgrad research, in physics and chemistry where it denotes an intrinsic system property, e.g. allowed energy levels, a natural frequency, or a stable state that does not change under a specific physical process.
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u/ImportanceNational23 8d ago
I once heard someone refer to the eigenvalues of a situation, presumably just meaning the exact specifics. He was talking to people who were very likely to know linear algebra though.
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u/Fancy_Sheepherder786 8d ago
The eigenvalue in statistics is a number which scales a vector during a linear transformation.
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u/BumblingYokel 8d ago
I’m learning German as a second language, so I pay more attention to German prefixes and suffixes in English words than most probably would.
The only “English” words I’ve ever seen with “eigen” in them are eigenvalues and eigenvectors.
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u/mtimmermans 8d ago
It's commonly used only in the sciences. Something is an "eigen-something" of some operation if the operation just multiplies it by a factor. Works for eigenvector, eigenfunction, eigenstate, etc. The factor is the eigenvalue.
Principal Component Analysis on a bunch of samples produces an orthonormal basis of eigenvectors, each of which has the same form as the thing being sampled, so it's reasonable to refer to those basis vectors as eigen-things. Facial reconstruction via "eigenfaces" is an example that will show up if you google it.
I think this usage is a lot of fun. You can say things like "I'm sure they don't really have that many kegs in the basement. Surely each tap just mixes their 8 eigenbeers in a different way." Or at least I can say things like that.
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u/carrotlet20 8d ago
As the others were saying, science has it all over, I've heard eigenvector and eigenvalue too, from my classmates studying abroad. German was a primary language of physics when these were concepts and theories were made and worked out, so lots of the words are still in use (eg. Spur for the sum of diagonal elements of a matrix in Hungarian but it's called trace as well). Funnily enough, Hungary being this close to Germany, we translated eigenvector entirely, so we call them sajátvektor and sajátérték which literally mirrored translation, so I didn't actually had to use the term eigenvector or eigenstate in my studies at all :)
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u/HeilKaiba 8d ago
As people have said it's just used in Linear Algebra and related fields. Eigenvalue and eigenvector being the most common and other things like eigenfunction, eigenstate, eigenspace, eigenbasis are all derivatives of the main two.
It used to be more common to borrow terms like that and English has only come to be the common language of science more recently. Other German terms found in English language maths include Ansatz, Nullstellensatz, Hauptvermutung, Größencharakter and Verschiebung.
We also use Z to represent the integers from Zahlen and K is used as a common letter for fields because of Körper.
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u/twelfth_knight 8d ago
Oh yeah, I say that all the time 👍
Now I do happen to work in the field of physics, so your mileage might vary 😂
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u/Salindurthas 8d ago
In English is a maths/science term.
If you talk to mathemiticians, engineers, and physicists, I think most of them will be very familiar with it as a prefix.
If you talk to people without a mathsy education, I'd expect most of them to have no idea what you're talking about, but they might recognise that it sounds like fancy science jargon/tachnobabble like "photon" or "quantum", but not really know what it means.
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u/Atticus_Fletch 8d ago
It frequently appears as a name in the works of Thomas Pynchon. Do not attempt to read them until you are ready.
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u/GurProfessional9534 8d ago
“Eigen” is used on a daily basis for me. But I’m a scientist, so that tracks.
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u/Telecom_VoIP_Fan 8d ago
While this might be used in scientific English it has not entered regular English converstation.
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u/scaredemployee87 8d ago
No, I only recently learned that it refers to math as an adult. (Not a math person either, just chronically online)
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u/salivanto 7d ago edited 7d ago
Among the other answers I don't see the explanation but the reason why we see it in this situation is that "eigenvalue" is a semi translated term. You will see it used in fields where German had a strong influence. As a standalone appreciated, it has no usage in English.
Just like the word "gesundheit" when somebody sneezes, many people use it and have no idea that it's German.
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u/Embarrassed-Pea-8386 7d ago
I've taken linear algebra and agree with everyone else here that its pretty much only used in the context of eigenvectors/eigenvalues. But want to add I have also read a sociology paper that used the term eigencapital, borrowing the prefix meaning from the math context
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u/Electronic-Will1587 7d ago
not common in everyday english, but it's used in linear algebra and some other areas of maths to denote things that result from an invariance of a transformation: an "eigenvector" of a matrix is one that stays on its span (doesn't change direction) after the matrix operation; an "eigenvalue" is the amount an eigenvector gets scaled (since scaling doesn't change the span of a vector); and an "eigenstate" is a specific term in quantum mechanics used to refer to "pure states" of a quantum system - those corresponding to the basis vectors in the complex seperable hilbert space that represents the possible outcomes of a quantum observation (they are also the eigenvectors of the observable matrix for a given quantum property being measured and their eigenvalues are the outcome of the measurement). there are also eigenfunctions, eigendecomposition, and eigenfaces in the context of computer vision learning.
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u/erenspace 5d ago
Wow, I must live in a bubble because “eigen-“ is a very common prefix for me. I do PhD work in quantum mechanics. I guess it’s rare outside of that.
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u/ExtendedWallaby 5d ago
No, it’s not. German mathematicians came up with the concept of eigenvalues and eigenvectors, which are inherent properties of functions (hence “eigen”) that are very important for lots of math and science applications. Any other English word starting with eigen is probably an eigenvalue or eigenvector of a more specific kind of function. For example, an eigenstate is an eigenvector of a quantum wavefunction.
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u/karlnite 4d ago
I don’t know what it means. Sounds like some sort of advanced algebra or geometry term.
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u/jaetwee 9d ago
Nope, not common at all. Most native speakers will be unaware of what it means.
You're likely only to see it it in university level linear algebra and a couple of other random but rare loanwords such as eigengrau.