Math 2.0

Linear and multilinear algebra; matrix theory Definition

Eigenspace

Builds on EigenvectorKernel of a linear map

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 eigenspace of TT for λ\lambda is Eλ=ker⁡(T−λid⁡V)={v∈V:T(v)=λv}.E_\lambda = \ker(T - \lambda \id_V) = \{v \in V : T(v) = \lambda v\}.

Remarks

The eigenspace is a subspace of VV; it is the kernel of a linear map. Its nonzero elements are the eigenvectors of TT for λ\lambda. Eλ≠0E_\lambda \ne 0 if and only if λ\lambda is an eigenvalue of TT. The dimension of EλE_\lambda is the geometric multiplicity of λ\lambda.

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

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Vector space, definition, no evidence levelVector spacedefinitionLinear map, definition, no evidence levelLinear mapdefinitionEndomorphism, definition, no evidence levelEndomorphismdefinitionKernel of a linear map, definition, no evidence levelKernel of a linear mapdefinitionEigenvalue, definition, no evidence levelEigenvaluedefinitionEigenvector, definition, no evidence levelEigenvectordefinitionEigenspace, definition, no evidence levelEigenspacedefinition
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. Endomorphism definition
  4. Kernel of a linear map definition
  5. Eigenvalue definition
  6. Eigenvector definition
  7. Eigenspace definition

Where this leads

2 pages build on this one.

Further reading

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