Eigenvalue

Definitions

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 transformation A is equal to the image of x under multiplication by λ; i.e. Ax=λx.

Equivalents

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.”

“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.”

“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.”

“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 […]”

CEFR level