eigenvalue

plural eigenvalues

1. linear algebraA scalar λ, such that there exists a non-zero vector x (a corresponding eigenvector) for which the image of x under a given linear transformation A is equal to the image of x under multiplication by λ; i.e. Ax=λx.

   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 […]

   2025 March 22 (last accessed), “Lesson 12”, in University College London, archived from the original on 21 Mar 2025, page 1

   3 more examples

   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.

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

   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.

   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

   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.

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

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