Math 2.0

Linear and multilinear algebra; matrix theory Eigenvalues and eigenvectors Theorem

Eigenvectors with distinct eigenvalues are linearly independent

Builds on EigenvectorLinear independence

Statement

Let FF be a field, let VV be a vector space over FF, and let T ⁣:V→VT \colon V \to V be a linear map. Let v1,…,vkv_1, \dots, v_k be eigenvectors of TT for eigenvalues λ1,…,λk\lambda_1, \dots, \lambda_k that are pairwise distinct. Then v1,…,vkv_1, \dots, v_k are linearly independent.

Remarks

The proof is by induction on kk: apply T−λkid⁡VT - \lambda_k \id_V to a linear relation, and the term of vkv_k disappears. So the sum of the eigenspaces of TT for distinct eigenvalues is a direct sum. In particular, an endomorphism of an nn-dimensional space has at most nn distinct eigenvalues.

Reading path

To understand this page from the ground up, read these 8 pages in order. The foundations come first.

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Vector space, definition, no evidence levelVector spacedefinitionLinear combination, definition, no evidence levelLinear combinationdefinitionLinear map, definition, no evidence levelLinear mapdefinitionLinear independence, definition, no evidence levelLinear independencedefinitionEndomorphism, definition, no evidence levelEndomorphismdefinitionEigenvalue, definition, no evidence levelEigenvaluedefinitionEigenvector, definition, no evidence levelEigenvectordefinitionEigenvectors with distinct eigenvalues are linearly independent, statement, no evidence levelEigenvectors with dis…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 combination definition
  3. Linear map definition
  4. Linear independence definition
  5. Endomorphism definition
  6. Eigenvalue definition
  7. Eigenvector definition
  8. Eigenvectors with distinct eigenvalues are linearly independent statement

Where this leads

1 page builds 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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Tag0HNMFG8F8K
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Kindstatement, theorem
WritingW0 stub
MSC15A18
StatusThis copy of the site has no build data. Run m2 status 0HNMFG8F8K.
UsesEigenvector, Linear independence

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