Math 2.0

Linear and multilinear algebra; matrix theory Definition

Geometric multiplicity

Builds on EigenspaceDimension

Definition

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. Let λ\lambda be an eigenvalue of TT. The geometric multiplicity of λ\lambda is dim⁡Eλ\dim E_\lambda, the dimension of the eigenspace Eλ=ker⁡(T−λid⁡V)E_\lambda = \ker(T - \lambda \id_V).

Remarks

It is the maximal number of linearly independent eigenvectors for λ\lambda. It satisfies 1≤dim⁡Eλ≤m1 \le \dim E_\lambda \le m, where mm is the algebraic multiplicity of λ\lambda. For (1101)\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}, the eigenvalue 11 has geometric multiplicity 11 and algebraic multiplicity 22. A matrix is diagonalisable if and only if pAp_A splits and the two multiplicities agree for each eigenvalue.

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Vector space, definition, no evidence levelVector spacedefinitionLinear combination, definition, no evidence levelLinear combinationdefinitionLinear map, definition, no evidence levelLinear mapdefinitionLinear subspace, definition, no evidence levelLinear subspacedefinitionLinear independence, definition, no evidence levelLinear independencedefinitionEndomorphism, definition, no evidence levelEndomorphismdefinitionKernel of a linear map, definition, no evidence levelKernel of a linear mapdefinitionLinear span, definition, no evidence levelLinear spandefinitionEigenvalue, definition, no evidence levelEigenvaluedefinitionSpanning set, definition, no evidence levelSpanning setdefinitionEigenvector, definition, no evidence levelEigenvectordefinitionBasis, definition, no evidence levelBasisdefinitionFinite-dimensional vector space, definition, no evidence levelFinite-dimensional ve…definitionEigenspace, definition, no evidence levelEigenspacedefinitionDimension, definition, no evidence levelDimensiondefinitionGeometric multiplicity, definition, no evidence levelGeometric multiplicitydefinition
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 combination definition
  3. Linear map definition
  4. Linear subspace definition
  5. Linear independence definition
  6. Endomorphism definition
  7. Kernel of a linear map definition
  8. Linear span definition
  9. Eigenvalue definition
  10. Spanning set definition
  11. Eigenvector definition
  12. Basis definition
  13. Finite-dimensional vector space definition
  14. Eigenspace definition
  15. Dimension definition
  16. Geometric multiplicity definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors, Eigenspaces, geometric multiplicity, and the eigenbasis
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