eigenvector

English

Etymology

From eigen- +‎ vector, a partial calque of German Eigenvektor.

The prefix eigen- (also used in eigenvalue) was first used in 1904, by David Hilbert, and was possibly inspired by a related usage by Hermann von Helmholtz.

Pronunciation

• Audio (US):

  (file)

Noun

eigenvector (plural eigenvectors)

1. (linear algebra) A vector that is only scaled (not rotated out of its span) under a particular linear transformation; a left or right eigenvector depending on context; (more formally) given a linear transformation A, a vector x such that Ax=λx [or xA=λx] for some scalar λ (called the eigenvalue).

   Synonyms: characteristic vector, latent vector, proper vector

   • 2025 March 22 (last accessed), “Lesson 11”, in University College London‎^[1], archived from the original on 6 October 2021, 8.2 Method to find eigenvalues and eigenvectors, page 1:

     Both equations give the relation x₁ = −x₂. Therefore the set of eigenvectors corresponding to λ = 2 is given by: […] An eigenvector corresponding to λ₁ = 2 is ( −1 1 ).

2. (physics, engineering) A right eigenvector; given a matrix A, the eigenvector of the transformation "left-side multiplication by A."

Usage notes

• Eigenvector has become the standard term in English, but proper vector was formerly more common.
• The concepts of eigenvector and eigenvalue arose from the study of quadratic forms and differential equations. Nowadays, they are normally introduced in the context of linear algebra.
• A linear transformation can always be represented as a matrix if the vector space is finite-dimensional (usually a safe assumption in physics).
• In consequence of the rules of matrix multiplication, left eigenvectors are row vectors, while right eigenvectors are column vectors. The convention of right eigenvector as "standard" is fundamentally an arbitrary choice.

Derived terms

• eigenbivector
• eigenvectorial
• left eigenvector
• right eigenvector

Related terms

• eigenbasis
• eigenbrain
• eigendecomposition
• eigenface
• eigenfunction
• eigenhead
• eigenmode
• eigenpair
• eigenspace
• eigenspinor
• eigenstate
• eigensystem
• eigentheory
• eigenvalue

Translations

vector not rotated by linear transformation

• Asturian: vector propiu m, autovector m
• Catalan: vector propi m
• Chinese:

  Mandarin: 本徵向量 / 本征向量 (zh) (běnzhēng xiàngliàng), 特徵向量 / 特征向量 (zh) (tèzhēng xiàngliàng)

• Danish: egenvektor c
• Dutch: eigenvector (nl) m
• Estonian: omavektor
• Finnish: ominaisvektori
• French: vecteur propre (fr) m
• Galician: vector propio m, autovector m, eigenvector m
• German: Eigenvektor (de) m
• Greek: ιδιοδιάνυσμα (el) n (idiodiánysma)
• Hebrew: וֶקְטוֹר עַצְמִי m
• Hindi: अभिलक्षणिक सदिश (abhilakṣaṇik sadiś)
• Hungarian: sajátvektor (hu)
• Icelandic: eiginvigur m, eiginvektor m
• Italian: autovettore (it) m
• Japanese: 固有ベクトル (ja) (koyū bekutoru)
• Norwegian:

  Bokmål: egenvektor m

  Nynorsk: eigenvektor m

• Polish: wektor własny (pl) m inan
• Portuguese: autovetor m, vetor próprio m
• Romanian: vector propriu m
• Russian: со́бственный ве́ктор m (sóbstvennyj véktor)
• Spanish: vector propio (es) m, autovector (es) m, eigenvector (es) m
• Swedish: egenvektor (sv) c
• Welsh: fector eigen m

Further reading

• eigenvector on Wikipedia.Wikipedia

See also

• Mathworld article on eigenvectors

Dutch

Etymology

From eigen +‎ vector, calque of German Eigenvektor.

Pronunciation

• IPA^(key): /ˈɛi̯.ɣənˌvɛk.tɔr/

• Audio:

  (file)

• Hyphenation: ei‧gen‧vec‧tor

Noun

eigenvector m (plural eigenvectoren, no diminutive)

1. (linear algebra) eigenvector

Spanish

Noun

eigenvector m (plural eigenvectores)

1. (linear algebra) eigenvector

   Synonyms: autovector, vector propio