Math 2.0

statement theorem

Existence of an eigenvalue over an algebraically closed field

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.

Explanations

L1 human

Over the complex numbers, every square matrix has at least one eigenvector. The reason is that every nonconstant complex polynomial has a root. Over the real numbers, a rotation can move every line, so no real eigenvector exists.

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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 node uses the nodes above it. The colour of the bar on a box shows the evidence level, and the box also gives it as text. The list below has the same nodes.
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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

Sources

  1. Web https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors, Eigenvalues and characteristic polynomial

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