Definitions For Eigenvector

Noun

EIGENVECTOR (plural EIGENVECTORs) (linear algebra) A vector that is not rotated under a particular linear transformation; a left eigenvector, 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, en, characteristic vector, latent vector, proper vector}} (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

eigenvectorial eigenbivector left eigenvector right eigenvector

Related terms

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

Translations

Chinese: : Mandarin: 本徵向量, tr=běnzhēng xiàngliàng, 特徵向量, tr=tèzhēng xiàngliàng Danish: egenvektor, c Dutch: eigenvector, m Estonian: omavektor Finnish: ominaisvektori French: vecteur propre, m German: Eigenvektor, m Greek: ιδιοδιάνυσμα, n Hebrew: וֶקְטוֹר עַצְמִי, m, sc=Hebr Icelandic: eiginvigur, m, eiginvektor, m Italian: autovettore, m Japanese: 固有ベクトル, tr=koyū bekutoru Norwegian: : Bokmål: egenvektor, m : Nynorsk: eigenvektor, m Polish: wektor własny, m Portuguese: autovetor, m, vetor próprio, m Romanian: vector propriu, m Russian: со́бственный ве́ктор, m Spanish: vector propio, m, autovector, m Swedish: egenvektor, c

Etymology

eigen + vector, a {{partial translation + en + de + nocap=1 + eigenvektor}}. The prefix eigen- (also used in eigenvalue) was first used in 1904, by {{w, David Hilbert}}, and was possibly inspired by a related usage by {{w, Hermann von Helmholtz}}.