Dictionary

eigenvalue en · NOUN

Pronunciation

• (US) audio
• /ˈaɪ.ɡənˌvæl.juː/ (Received-Pronunciation)
• /ˈaɪ.ɡənˌvæl.ju/ (General-American)

Etymology

From eigen- + value. Partial calque of German Eigenwert.

Meanings

1. A scalar λ, such that there exists a non-zero vector x (a corresponding eigenvector) for which the image of x under a given linear transformation A is equal to the image of x under multiplication by λ; i.e. Ax=λx.
   • For many quantum-mechanical problems it is important to investigate the change of eigenvalues and eigenfunctions with the continuous change of one or more parameters. The case in which one knows the eigenvalues and eigenfunctions for two special values of the parameters, and is interested in the region in between is particularly interesting. 2000, John von Neumann, E. Wigner, “On the Behaviour of Eigenvalues in Adiabatic Processes”, in Hinne Hettema, transl., edited by Hinne Hettema, Quantum Chemistry: Classic Scientific Papers, World Scientific, published 1929, page 25:
   • Problems that require an investigation of eigenvalues and eigenfunctions arise in connection with numerous topics in mechanics, the theory of vibrations and stability, hydrodynamics, elasticity, acoustics, electrodynamics, quantum mechanics, etc. 2005, Leonid D. Akulenko, Sergei V. Nesterov, High-Precision Methods in Eigenvalue Problems and Their Applications, CRC Press (Chapman & Hall), page 1:
   • In the extension, one associates eigenvalues, sets of scalars, with arrays of matrices by considering the singularity of linear combinations of the matrices in the various rows, involving the same coefficients in each case. Attention to this area was called in the early 1920's by R. D. Carmichael, who pointed out in addition the enormous variety of mixed eigenvalue problems with several parameters. 1972, F. V. Atkinson, Multiparameter Eigenvalue Problems, Volume I: Matrices and Compact Operators, Academic Press, page x:
   • To find all the eigenvalues of A, solve the characteristic equation. ¶ For each eigenvalue λ, to find the corresponding set of eigenvectors, solve the linear system of equations […] 2025 March 22 (last accessed), “Lesson 12”, in University College London, archived from the original on 21 Mar 2025, page 1:

Forms

Deriveds

right eigenvalue · left eigenvalue

Synonyms

characteristic value · eigenroot · proper value · characteristic root · latent value

Translations (39)

es valor propio (scalar multiplier of an eigenvector) · pl wartość własna (scalar multiplier of an eigenvector) · da egenværdi (scalar multiplier of an eigenvector) · gl eigenvalor (scalar multiplier of an eigenvector) · ast valor propiu (scalar multiplier of an eigenvector) · sk vlastná hodnota (scalar multiplier of an eigenvector) · ja 固有値 (scalar multiplier of an eigenvector) · gl valor propio (scalar multiplier of an eigenvector) · nb egenverdi (scalar multiplier of an eigenvector) · cmn 特徵值 /特征值 (scalar multiplier of an eigenvector) · he עֵרֶךְ עַצְמִי (scalar multiplier of an eigenvector) · lt tikrinė reikšmė (scalar multiplier of an eigenvector) · hu sajátérték (scalar multiplier of an eigenvector) · cs vlastní hodnota (scalar multiplier of an eigenvector) · ro valoare proprie (scalar multiplier of an eigenvector) · fa مقدار ویژه (scalar multiplier of an eigenvector) · es autovalor (scalar multiplier of an eigenvector) · is eigingildi (scalar multiplier of an eigenvector) · it autovalore (scalar multiplier of an eigenvector) · ca valor propi (scalar multiplier of an eigenvector) · fi ominaisarvo (scalar multiplier of an eigenvector) · sv egenvärde (scalar multiplier of an eigenvector) · nl eigenwaarde (scalar multiplier of an eigenvector) · cy gwerth eigen (scalar multiplier of an eigenvector) · nn eigenverdi (scalar multiplier of an eigenvector) · et omaväärtus (scalar multiplier of an eigenvector) · hi अभिलाक्षणिक मान (scalar multiplier of an eigenvector) · ru со́бственное значе́ние (scalar multiplier of an eigenvector) · pt valor próprio (scalar multiplier of an eigenvector) · el ιδιοτιμή (scalar multiplier of an eigenvector) · pt valor característico (scalar multiplier of an eigenvector) · pt autovalor (scalar multiplier of an eigenvector) · de Eigenwert (scalar multiplier of an eigenvector) · fr valeur propre (scalar multiplier of an eigenvector) · es eigenvalor (scalar multiplier of an eigenvector) · es valor característico (scalar multiplier of an eigenvector) · gl autovalor (scalar multiplier of an eigenvector) · ast autovalor (scalar multiplier of an eigenvector) · cmn 本徵值 /本征值 (scalar multiplier of an eigenvector)