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EtymologyEdit

eigen- +‎ value

PronunciationEdit

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

NounEdit

eigenvalue (plural eigenvalues)

  1. (linear algebra) A scalar,  , such that there exists a vector   (the corresponding eigenvector) for which the image of   under a given linear operator   is equal to the image of   under multiplication by  ; i.e.  .
    The eigenvalues   of a square transformation matrix   may be found by solving  .
    • 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, Hinne Hettema (translator), J. Von Neumann, E. Wigner, On the Behaviour of Eigenvalues in Adiabatic Processes [1929], Hinne Hettema (editor), Quantum Chemistry: Classic Scientific Papers, World Scientific, 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 notesEdit

When unqualified, as in the above example, eigenvalue conventionally refers to a right eigenvalue, characterised by   for some right eigenvector  . Left eigenvalues, characterised by   also exist with associated left eigenvectors  . For commutative operators, the left eigenvalues and right eigenvalues will be the same, and are referred to as eigenvalues with no qualifier.

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