PlanetMath: multiplicity of eigenvalue

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multiplicity of eigenvalue

(Definition)

Suppose is a finite dimensional vector space over a field $\mathbbmss{F}$ , and suppose $L\colon V\to V$ is a linear map. Suppose also that $\lambda\in \mathbbmss{F}$ is an eigenvalue of , that is, $\operatorname{det}( L - \lambda I)=0$ .

The algebraic multiplicity, denoted by $A_\lambda(L)$ , of $\lambda$ is the multiplicity of the root $\lambda$ to the polynomial $\operatorname{det}( L - \lambda I)=0$ . The geometric multiplicity of $\lambda$ , denoted by $G_\lambda(L)$ , is the dimension of $\ker ( L - \lambda I)$ , the eigenspace of $\lambda$ .

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"multiplicity of eigenvalue" is owned by matte. [ full author list (2) ]

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Also defines:

geometric multiplicity, algebraic multiplicity

Keywords:

eigenvalue

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Attachments:: algebraic and geometric multiplicity do not coincide (Example) by matte

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Cross-references: eigenspace, dimension, polynomial, multiplicity of the root, eigenvalue, linear map, field, vector space, finite dimensional
There are 4 references to this entry.

This is version 2 of multiplicity of eigenvalue, born on 2005-05-10, modified 2007-07-08.
Object id is 7036, canonical name is MultiplicityOfEigenvalue.
Accessed 8905 times total.

Classification:

AMS MSC:

15A18 (Linear and multilinear algebra; matrix theory :: Eigenvalues, singular values, and eigenvectors)

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