eigenvalue

English

Etymology

eigen- +‎ value, a partial calque of German Eigenwert.

Pronunciation

• enPR: īʹgən'vălyo͞o, IPA^(key): /ˈaɪɡənˌvæljuː/

• Audio (US)

  (file)

Noun

eigenvalue (plural eigenvalues)

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

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

   • 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 l920'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, 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.

Usage notes

When unqualified, as in the above example, eigenvalue conventionally refers to a right eigenvalue, characterised by ${\displaystyle \mathrm {M} x=\lambda x}$ for some right eigenvector ${\displaystyle x}$. Left eigenvalues, characterised by ${\displaystyle y\mathrm {M} =y\lambda }$ also exist with associated left eigenvectors ${\displaystyle y}$. (In consequence of the equations, left eigenvectors are row vectors, while right eigenvectors are column vectors.) The convention of right eigenvector as "standard" is fundamentally an arbitrary choice.

Synonyms

• (scalar multiplier of an eigenvector): characteristic root, characteristic value, eigenroot, latent value, proper value

Translations

See also

• eigenbasis
• eigenbrain
• eigendecomposition, eigen decomposition
• eigenface
• eigenfunction
• eigenhead
• eigenmode
• eigenpair
• eigenspace
• eigenspinor
• eigenstate
• eigensystem
• eigentheory
• eigenvector

Further reading

• Eigenvalues and eigenvectors on Wikipedia.Wikipedia
• Transformation matrix on Wikipedia.Wikipedia
• Eigenvalue on MathWorld.
• Eigen value on Encyclopedia of Mathematics.