eigenvalue

English

Etymology

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

Pronunciation

• (Received Pronunciation) IPA^(key): /ˈaɪ.ɡənˌvæl.juː/
• (General American) IPA^(key): /ˈaɪ.ɡənˌvæl.ju/

• Audio (US):

  (file)

• Hyphenation: ei‧gen‧val‧ue

Noun

eigenvalue (plural eigenvalues)

1. (linear algebra) A scalar $\lambda$, such that there exists a non-zero vector $\mathrm{x}$ (a corresponding eigenvector) for which the image of $\mathrm{x}$ under a given linear transformation $A$ is equal to the image of $\mathrm{x}$ under multiplication by $\lambda$; i.e. $A\mathrm{x}=\lambda \mathrm{x}$.

   Synonyms: characteristic root, characteristic value, eigenroot, latent value, proper value

   Hyponyms: left eigenvalue, right eigenvalue

   • 1972, F. V. Atkinson, Multiparameter Eigenvalue Problems, Volume I: Matrices and Compact Operators, Academic Press, page x:

     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.

   • 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:

     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.

   • 2005, Leonid D. Akulenko, Sergei V. Nesterov, High-Precision Methods in Eigenvalue Problems and Their Applications‎^[1], CRC Press (Chapman & Hall), page 1:

     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.

   • 2025 March 22 (last accessed), “Lesson 12”, in University College London‎^[2], archived from the original on 21 March 2025, page 1:

     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 […]

Usage notes

• Eigenvalue is the standard term in English, displacing the older proper value.
• When unqualified, eigenvalue conventionally refers to a right eigenvalue, characterised by $M\mathrm{x}=\lambda \mathrm{x}$ for an associated right eigenvector $\mathrm{x}$. Left eigenvalues, characterised by $\mathrm{y}M=\mathrm{y}\lambda$ also exist with an associated left eigenvectors $\mathrm{y}$.

Derived terms

• left eigenvalue
• right eigenvalue

Translations

scalar multiplier of an eigenvector

• Asturian: valor propiu m, autovalor m
• Catalan: valor propi m
• Chinese:

  Mandarin: 特徵值 / 特征值 (zh) (tèzhēngzhí), 本徵值 / 本征值 (zh) (běnzhēngzhí)

• Czech: vlastní hodnota f
• Danish: egenværdi c
• Dutch: eigenwaarde (nl) f
• Estonian: omaväärtus
• Finnish: ominaisarvo
• French: valeur propre (fr) f
• Galician: valor propio m, autovalor m, eigenvalor m
• German: Eigenwert (de) m
• Greek: ιδιοτιμή (el) f (idiotimí)
• Hebrew: עֵרֶךְ עַצְמִי m
• Hindi: अभिलाक्षणिक मान (abhilākṣaṇik mān)
• Hungarian: sajátérték (hu)
• Icelandic: eigingildi n
• Italian: autovalore (it) m
• Japanese: 固有値 (ja) (koyūchi)
• Lithuanian: tikrinė reikšmė
• Norwegian:

  Bokmål: egenverdi m

  Nynorsk: eigenverdi m

• Persian: مقدار ویژه (meqdâr-e viže)
• Polish: wartość własna (pl) f
• Portuguese: (Brazil) autovalor m, (Portugal) valor próprio m, valor característico m
• Romanian: valoare proprie f
• Russian: со́бственное значе́ние n (sóbstvennoje značénije)
• Slovak: vlastná hodnota f
• Spanish: autovalor (es) m, valor propio (es) m, valor característico m, eigenvalor (es) m
• Swedish: egenvärde (sv) n
• Welsh: gwerth eigen m