Let V be a vector spaceMathworldPlanetmath over a field k, and let A be an endomorphism of V (meaning a linear mapping of V into itself). A scalar λk is said to be an eigenvalueMathworldPlanetmathPlanetmathPlanetmathPlanetmath of A if there is a nonzero xV for which

Ax=λx. (1)

Geometrically, one thinks of a vector whose direction is unchanged by the action of A, but whose magnitude is multiplied by λ.

If V is finite dimensional, elementary linear algebra shows that there are several equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath definitions of an eigenvalue:

(2) The linear mapping


i.e. B:xλx-Ax, has no inversePlanetmathPlanetmathPlanetmathPlanetmathPlanetmath.

(3) B is not injectivePlanetmathPlanetmath.

(4) B is not surjective.

(5) det(B)=0, i.e. det(λI-A)=0.

But if V is of infiniteMathworldPlanetmathPlanetmath dimensionPlanetmathPlanetmathPlanetmath, (5) has no meaning and the conditions (2) and (4) are not equivalent to (1). A scalar λ satisfying (2) (called a spectral value of A) need not be an eigenvalue. Consider for example the complex vector space V of all sequencesMathworldPlanetmath (xn)n=1 of complex numbersMathworldPlanetmathPlanetmath with the obvious operationsMathworldPlanetmath, and the map A:VV given by


Zero is a spectral value of A, but clearly not an eigenvalue.

Now suppose again that V is of finite dimension, say n. The function


is a polynomialPlanetmathPlanetmath of degree n over k in the variable λ, called the characteristic polynomialPlanetmathPlanetmath of the endomorphism A. (Note that some writers define the characteristic polynomial as det(A-λI) rather than det(λI-A), but the two have the same zeros.)

If k is or any other algebraically closed field, or if k= and n is odd, then χ has at least one zero, meaning that A has at least one eigenvalue. In no case does A have more than n eigenvalues.

Although we didn’t need to do so here, one can compute the coefficients of χ by introducing a basis of V and the corresponding matrix for B. Unfortunately, computing n×n determinantsMathworldPlanetmath and finding roots of polynomials of degree n are computationally messy procedures for even moderately large n, so for most practical purposes variations on this naive scheme are needed. See the eigenvalue problem for more information.

If k= but the coefficients of χ are real (and in particular if V has a basis for which the matrix of A has only real entries), then the non-real eigenvalues of A appear in conjugatePlanetmathPlanetmathPlanetmath pairs. For example, if n=2 and, for some basis, A has the matrix


then χ(λ)=λ2+1, with the two zeros ±i.

Eigenvalues are of relatively little importance in connection with an infinite-dimensional vector space, unless that space is endowed with some additional structureMathworldPlanetmath, typically that of a Banach spaceMathworldPlanetmath or Hilbert spaceMathworldPlanetmath. But in those cases the notion is of great value in physics, engineering, and mathematics proper. Look for “spectral theory” for more on that subject.

Title eigenvalue
Canonical name Eigenvalue
Date of creation 2013-03-22 12:11:52
Last modified on 2013-03-22 12:11:52
Owner Koro (127)
Last modified by Koro (127)
Numerical id 15
Author Koro (127)
Entry type Definition
Classification msc 15A18
Related topic EigenvalueProblem
Related topic SimilarMatrix
Related topic EigenvectorMathworldPlanetmathPlanetmathPlanetmath
Related topic SingularValueDecomposition
Defines eigenvalue
Defines spectral value