View source for Eigenvalue

==English==

===Etymology===
From {{prefix|en|eigen|value}}, a {{partial calque|en|de|nocap=1|Eigenwert}}.

===Pronunciation===
* {{enPR|īʹgən'vălyo͞o}}, {{IPA|en|/ˈaɪɡənˌvæljuː/}}
* {{audio|en|en-us-eigenvalue.ogg|a=US}}

===Noun===
{{en-noun}}

# {{lb|en|linear algebra}} A [[scalar]], <math>\lambda</math>, such that there exists a non-zero [[vector]] <math>x</math> (a corresponding [[eigenvector]]) for which the [[image]] of <math>x</math> under a given [[linear operator]] <math>\mathrm{A}</math> is equal to the [[image]] of <math>x</math> under [[multiplication]] by <math>\lambda</math>; i.e. <math>\mathrm{A} x = \lambda x</math>.
#: {{ux|en|The '''eigenvalues''' <math>\lambda</math> of a square transformation matrix <math>\mathrm{M}</math> may be found by solving <math>\det(\mathrm{M} - \lambda\mathrm{I}) = 0</math>.}}
#* {{quote-book|en|year=1972|author={{w|Frederick Valentine Atkinson|F. V. Atkinson}}|title=Multiparameter '''Eigenvalue''' Problems, Volume I: Matrices and Compact Operators|pageurl=https://books.google.com.au/books?id=VRYSzh33uPMC&pg=PR10&dq=%22eigenvalue%22%7C%22eigenvalues%22&hl=en&sa=X&ved=0ahUKEwjP97SwuO_XAhUBG5AKHbVzCpAQ6AEI4gEwIQ#v=onepage&q=%22eigenvalue%22%7C%22eigenvalues%22&f=false|page=x|publisher=Academic Press
|passage=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.}}
#* {{quote-book|en|year=2000|translator=Hinne Hettema|author=w:John von Neumann; {{w|Eugene Wigner|E. Wigner}}|chapter=On the Behaviour of '''Eigenvalues''' in Adiabatic Processes|year_published=1929|editor=Hinne Hettema|title=Quantum Chemistry: Classic Scientific Papers|publisher=World Scientific|pageurl=https://books.google.com.au/books?id=qsidHRJmUoIC&pg=PA25&dq=%22eigenvalue%22%7C%22eigenvalues%22&hl=en&sa=X&ved=0ahUKEwjP97SwuO_XAhUBG5AKHbVzCpAQ6AEI1wIwNg#v=onepage&q=%22eigenvalue%22%7C%22eigenvalues%22&f=false|page=25
|passage=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.}}
#* {{quote-book|en|year=2005|author=Leonid D. Akulenko; Sergei V. Nesterov|title=High-Precision Methods in '''Eigenvalue''' Problems and Their Applications|publisher=CRC Press (Chapman & Hall)|url=https://books.google.com.au/books?id=kKjMSwgH-noC&pg=PA1&dq=%22eigenvalue%22%7C%22eigenvalues%22&hl=en&sa=X&ved=0ahUKEwjP97SwuO_XAhUBG5AKHbVzCpAQ6AEIwgEwHA#v=onepage&q=%22eigenvalue%22%7C%22eigenvalues%22&f=false|page=1
|passage=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.}}

====Usage notes====
* When unqualified, as in the above example, '''eigenvalue''' conventionally refers to a [[right eigenvalue|''right'' eigenvalue]], characterised by <math>\mathrm{M} x = \lambda x</math> for some [[right eigenvector]] <math>x</math>. [[left eigenvalue|''Left'' eigenvalues]], characterised by <math>y \mathrm{M} = y\lambda</math> also exist with associated [[left eigenvector]]s <math>y</math>. (In consequence of the equations, left eigenvectors are [[row vector]]s, while right eigenvectors are [[column vector]]s.) The convention of right eigenvector as "standard" is fundamentally an arbitrary choice.

====Synonyms====
* {{sense|scalar multiplier of an eigenvector}} {{l|en|characteristic root}}, {{l|en|characteristic value}}, {{l|en|eigenroot}}, {{l|en|latent value}}, {{l|en|proper value}}

====Translations====
{{trans-top|scalar multiplier of an eigenvector}}
* Chinese:
*: Mandarin: {{t+|cmn|特徵值|tr=tèzhēngzhí}}, {{t+|cmn|本徵值|tr=běnzhēngzhí}}
* Czech: {{t|cs|vlastní hodnota|f}}
* Danish: {{t|da|egenværdi|c}}
* Dutch: {{t+|nl|eigenwaarde|f}}
* Estonian: {{t|et|omaväärtus}}
* Finnish: {{t|fi|ominaisarvo}}
* French: {{t+|fr|valeur propre|f}}
* German: {{t+|de|Eigenwert|m}}
* Greek: {{t+|el|ιδιοτιμή|f}}
* Hebrew: {{t|he|עֵרֶךְ עַצְמִי|m|sc=Hebr}}
* Hindi: {{t|hi|अभिलाक्षणिक मान}}
* Icelandic: {{t|is|eigingildi|n}}
* Italian: {{t+|it|autovalore|m}}
* Japanese: {{t+|ja|固有値|tr=koyūchi}}
* Lithuanian: {{t|lt|tikrinė reikšmė}}
* Norwegian:
*: Bokmål: {{t|nb|egenverdi|m}}
*: Nynorsk: {{t|nn|eigenverdi|m}}
* Persian: {{t|fa|مقدار ویژه|tr=meqdâr-e viže}}
* Polish: {{t+|pl|wartość własna|f}}
* Portuguese: {{t|pt|valor próprio|m}} {{q|Portugal}}, {{t|pt|autovalor|m}} {{q|Brazil}}
* Romanian: {{t|ro|valoare proprie|f}}
* Russian: {{t|ru|со́бственное значе́ние|n}}
* Slovak: {{t|sk|vlastná hodnota|f}}
* Spanish: {{t|es|autovalor|m}}, {{t|es|valor propio|m}}, [[valor característico]] {{g|m}} {{qualifier|dated}}
* Swedish: {{t+|sv|egenvärde|n}}
* Welsh: {{t|cy|gwerth eigen|m}}
{{trans-bottom}}

===See also===
* {{l|en|eigenbasis}}
* {{l|en|eigenbrain}}
* {{l|en|eigendecomposition}}, {{l|en|eigen decomposition}}
* {{l|en|eigenface}}
* {{l|en|eigenfunction}}
* {{l|en|eigenhead}}
* {{l|en|eigenmode}}
* {{l|en|eigenpair}}
* {{l|en|eigenspace}}
* {{l|en|eigenspinor}}
* {{l|en|eigenstate}}
* {{l|en|eigensystem}}
* {{l|en|eigentheory}}
* {{l|en|eigenvector}}

===Further reading===
* {{pedia|Eigenvalues and eigenvectors}}
* {{pedia|Transformation matrix}}
* [http://mathworld.wolfram.com/Eigenvalue.html Eigenvalue] on {{w|MathWorld}}.
* [https://www.encyclopediaofmath.org/index.php/Eigen_value Eigen value] on {{w|Encyclopedia of Mathematics}}.

{{cln|en|hybridisms}}