eigenvalue

Etymology

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

noun

1. (linear algebra) A scalar, λ, such that there exists a non-zero vector x (a corresponding eigenvector) for which the image of x under a given linear operator A is equal to the image of x under multiplication by λ; i.e. Ax=λx.
   The eigenvalues λ of a square transformation matrix M may be found by solving det (M-λI)=0.
   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. 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