Math 2.0

Linear and multilinear algebra; matrix theory Definition

Generalised eigenspace

Builds on Generalised eigenvector

Definition

Let FF be a field, let VV be a vector space over FF, let T ⁣:V→VT \colon V \to V be an endomorphism, and let λ∈F\lambda \in F. The generalised eigenspace of TT for λ\lambda is Vλ={v∈V:(T−λid⁡V)k(v)=0 for some k≥1}.V_\lambda = \{v \in V : (T - \lambda \id_V)^k (v) = 0 \text{ for some } k \ge 1\}.

Remarks

It is a TT-invariant subspace that contains the eigenspace EλE_\lambda. If dim⁡V=n\dim V = n is finite, then Vλ=ker⁡(T−λid⁡V)nV_\lambda = \ker (T - \lambda \id_V)^n. Its dimension is the algebraic multiplicity of λ\lambda, in finite dimensions. Its nonzero elements are the generalised eigenvectors for λ\lambda.

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Vector space, definition, no evidence levelVector spacedefinitionLinear map, definition, no evidence levelLinear mapdefinitionEndomorphism, definition, no evidence levelEndomorphismdefinitionEigenvalue, definition, no evidence levelEigenvaluedefinitionPolynomial of an endomorphism, definition, no evidence levelPolynomial of an endo…definitionEigenvector, definition, no evidence levelEigenvectordefinitionGeneralised eigenvector, definition, no evidence levelGeneralised eigenvect…definitionGeneralised eigenspace, definition, no evidence levelGeneralised eigenspacedefinition
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  1. Vector space definition
  2. Linear map definition
  3. Endomorphism definition
  4. Eigenvalue definition
  5. Polynomial of an endomorphism definition
  6. Eigenvector definition
  7. Generalised eigenvector definition
  8. Generalised eigenspace definition

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Further reading

  1. Web https://en.wikipedia.org/wiki/Generalized_eigenvector, Overview and definition
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