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[parent] multiplicity of eigenvalue (Definition)

Suppose $ V$ 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 $ L$, 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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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
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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.

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AMS MSC15A18 (Linear and multilinear algebra; matrix theory :: Eigenvalues, singular values, and eigenvectors)

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