Math 2.0

Linear and multilinear algebra; matrix theory Definition

Generalised eigenvector

Builds on EigenvectorPolynomial of an endomorphism

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. A generalised eigenvector of TT for λ\lambda is a vector v∈Vv \in V with v≠0v \ne 0 and (T−λid⁡V)k(v)=0(T - \lambda \id_V)^k (v) = 0 for some integer k≥1k \ge 1. The least such kk is the rank of the generalised eigenvector.

Remarks

The generalised eigenvectors of rank 11 are the eigenvectors. If a generalised eigenvector for λ\lambda exists, then λ\lambda is an eigenvalue. For A=(1101)A = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}, the vector e2e_2 is a generalised eigenvector for 11 of rank 22, and it is not an eigenvector.

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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…definition
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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

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Further reading

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