Definition of "eigenvalue" (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 . A x = λ x . quotations examples
Quotations 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 l 920'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