Math 2.0

Linear and multilinear algebra; matrix theory Eigenvalues and eigenvectors Theorem

Existence of an eigenvalue over an algebraically closed field

Builds on EigenvalueCharacteristic polynomial of a linear map

Statement

Let FF be an algebraically closed field, let VV be a finite-dimensional vector space over FF with V≠0V \ne 0, and let T ⁣:V→VT \colon V \to V be a linear map. Then TT has at least one eigenvalue in FF.

Remarks

The characteristic polynomial of TT has degree dim⁡V≥1\dim V \ge 1, so it has a root in FF. The statement applies to F=ℂF = \C, by the fundamental theorem of algebra. It fails over ℝ\R: the rotation of ℝ2\R^2 by 9090 degrees has no real eigenvalue. It also fails in infinite dimensions: the map p(x)↦x p(x)p(x) \mapsto x\, p(x) on ℂ[x]\C[x] has no eigenvalue.

Reading path

To understand this page from the ground up, read these 21 pages in order. The foundations come first.

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Identity matrix, object, no evidence levelIdentity matrixobjectMatrix, definition, no evidence levelMatrixdefinitionVector space, definition, no evidence levelVector spacedefinitionSquare matrix, definition, no evidence levelSquare matrixdefinitionLinear 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 levelEndomorphismdefinitionLinear span, definition, no evidence levelLinear spandefinitionMultilinear map, definition, no evidence levelMultilinear mapdefinitionEigenvalue, definition, no evidence levelEigenvaluedefinitionSpanning set, definition, no evidence levelSpanning setdefinitionAlternating multilinear map, definition, no evidence levelAlternating multiline…definitionBasis, definition, no evidence levelBasisdefinitionDeterminant, definition, no evidence levelDeterminantdefinitionCoordinate vector, definition, no evidence levelCoordinate vectordefinitionCharacteristic polynomial, definition, no evidence levelCharacteristic polyno…definitionMatrix of a linear map, definition, no evidence levelMatrix of a linear mapdefinitionCharacteristic polynomial of a linear map, definition, no evidence levelCharacteristic polyno…definitionExistence of an eigenvalue over an algebraically closed field, statement, no evidence levelExistence of an eigen…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. Identity matrix object
  2. Matrix definition
  3. Vector space definition
  4. Square matrix definition
  5. Linear combination definition
  6. Linear map definition
  7. Linear subspace definition
  8. Linear independence definition
  9. Endomorphism definition
  10. Linear span definition
  11. Multilinear map definition
  12. Eigenvalue definition
  13. Spanning set definition
  14. Alternating multilinear map definition
  15. Basis definition
  16. Determinant definition
  17. Coordinate vector definition
  18. Characteristic polynomial definition
  19. Matrix of a linear map definition
  20. Characteristic polynomial of a linear map definition
  21. Existence of an eigenvalue over an algebraically closed field statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors, Eigenvalues and characteristic polynomial
About this page
Tag08CJB2NQ92
URIm2://08CJB2NQ92
Kindstatement, theorem
WritingW0 stub
MSC15A18
StatusThis copy of the site has no build data. Run m2 status 08CJB2NQ92.
UsesEigenvalue, Characteristic polynomial of a linear map

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