eigenvalue

English

Etymology

eigen- +‎ value

Pronunciation

• enPR: īʹgən'vălyo͞o, IPA: /ˈaɪɡənˌvæljuː/

• Audio (US)

  (file)

Noun

eigenvalue (plural eigenvalues)

1. (linear algebra) The change in magnitude of a vector that does not change in direction under a given linear transformation; a scalar factor by which an eigenvector is multiplied under such a transformation.

   The eigenvalues $\lambda\!$ of a transformation matrix $\rm M\!$ may be found by solving $\det({\rm M} - \lambda {\rm I}) = 0\!$ .

Usage notes

When unqualified, as in the above example, eigenvalue conventionally refers to a right eigenvalue, charactarised by ${\rm M} x = \lambda x\!$ for some right eigenvector $x\!$. Left eigenvalues, charactarised by $y {\rm M} = y \lambda\!$ also exist with associated left eigenvectors $y\!$. For commutative operators, the left eigenvalues and right eigenvalues will be the same, and are referred to as eigenvalues with no qualifier.

Synonyms

• characteristic root
• characteristic value
• eigenroot
• latent value

Related terms

• eigen decomposition
• eigenfunction
• eigenvector

Translations

change in magnitude of an eigenvector

• Czech: vlastní hodnota f
• Danish: egenværdi (da) c
• Dutch: eigenwaarde (nl) f
• Estonian: omaväärtus (et)
• Finnish: ominaisarvo (fi)
• French: valeur propre f
• German: Eigenwert (de) m
• Greek: ιδιοτιμή (el) (idiotimí) f 
• Icelandic: eigingildi (is) n
• Italian: autovalore (it) m
• Japanese: 固有値 (koyūchi)

• Lithuanian: tikrinė reikšmė
• Norwegian:

  Bokmål: egenverdi (no) m

  Nynorsk: eigenverdi (nn) m

• Polish: wartość własna (pl) f
• Portuguese: valor próprio m
• Romanian: valoare proprie f
• Russian: собственное значение (sóbstvennoje značénije) n
• Slovak: vlastná hodnota f
• Spanish: autovalor m, valor propio m, valor característico m (dated)
• Swedish: egenvärde (sv) n

See also

• Eigenvalue on Wikipedia.Wikipedia
• Mathworld article on eigenvalues