Math 2.0

Linear and multilinear algebra; matrix theory Canonical forms and decompositions Theorem

Decomposition into generalised eigenspaces

Builds on Generalised eigenspaceDirect sum of subspaces

Statement

Let FF be a field, let VV be a finite-dimensional vector space over FF, and let T ⁣:V→VT \colon V \to V be a linear map. Assume that the characteristic polynomial of TT splits over FF as pT(t)=(t−λ1)m1⋯(t−λk)mkp_T(t) = (t - \lambda_1)^{m_1} \cdots (t - \lambda_k)^{m_k} with pairwise distinct λi\lambda_i. Then V=Vλ1⊕⋯⊕Vλk,V = V_{\lambda_1} \oplus \cdots \oplus V_{\lambda_k}, where VλiV_{\lambda_i} is the generalised eigenspace of λi\lambda_i, and dim⁡Vλi=mi\dim V_{\lambda_i} = m_i.

Remarks

The hypothesis holds for every TT when FF is algebraically closed, for example F=ℂF = \C. Each VλiV_{\lambda_i} is TT-invariant, and T−λiid⁡T - \lambda_i \id is nilpotent on it. A proof uses the coprime polynomials (t−λi)mi(t - \lambda_i)^{m_i} and the Cayley–Hamilton theorem. This direct sum decomposition is the first step towards the Jordan normal form.

Reading path

To understand this page from the ground up, read these 12 pages in order. The foundations come first.

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Vector space, definition, no evidence levelVector spacedefinitionLinear map, definition, no evidence levelLinear mapdefinitionLinear subspace, definition, no evidence levelLinear subspacedefinitionEndomorphism, definition, no evidence levelEndomorphismdefinitionSum of subspaces, definition, no evidence levelSum of subspacesdefinitionEigenvalue, definition, no evidence levelEigenvaluedefinitionPolynomial of an endomorphism, definition, no evidence levelPolynomial of an endo…definitionDirect sum of subspaces, definition, no evidence levelDirect sum of subspac…definitionEigenvector, definition, no evidence levelEigenvectordefinitionGeneralised eigenvector, definition, no evidence levelGeneralised eigenvect…definitionGeneralised eigenspace, definition, no evidence levelGeneralised eigenspacedefinitionDecomposition into generalised eigenspaces, statement, no evidence levelDecomposition into ge…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Vector space definition
  2. Linear map definition
  3. Linear subspace definition
  4. Endomorphism definition
  5. Sum of subspaces definition
  6. Eigenvalue definition
  7. Polynomial of an endomorphism definition
  8. Direct sum of subspaces definition
  9. Eigenvector definition
  10. Generalised eigenvector definition
  11. Generalised eigenspace definition
  12. Decomposition into generalised eigenspaces statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Generalized_eigenvector, Jordan normal form
About this page
Tag0Z3GDY89T5
URIm2://0Z3GDY89T5
Kindstatement, theorem
WritingW0 stub
MSC15A18, 15A21
StatusThis copy of the site has no build data. Run m2 status 0Z3GDY89T5.
UsesGeneralised eigenspace, Direct sum of subspaces

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