eigenvector

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.

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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

   • (Can we date this quote?), “Lesson 11”, in University College London‎^[1], 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."

• 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.

• eigenvectorial
• eigenbivector
• left eigenvector
• right eigenvector

• Mathworld article on eigenvectors

From eigen +‎ vector, calque of German Eigenvektor.

• IPA^(key): /ˈɛi̯.ɣənˌvɛk.tɔr/

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• Hyphenation: ei‧gen‧vec‧tor

eigenvector m (plural eigenvectoren)

1. (linear algebra) eigenvector

eigenvector m (plural eigenvectores)

1. (linear algebra) eigenvector

   Synonyms: autovector, vector propio