Math 2.0

statement theorem

Cayley–Hamilton theorem for linear maps

Statement

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 with characteristic polynomial pTp_T. Then pT(T)=0p_T(T) = 0, where pT(T)p_T(T) is the value of pTp_T at TT.

Remarks

It follows from the Cayley–Hamilton theorem for matrices applied to the matrix of TT in a basis. The map p↦p([T]B)p \mapsto p([T]_B) corresponds to p↦p(T)p \mapsto p(T) under T↦[T]BT \mapsto [T]_B. The statement also holds for endomorphisms of finitely generated modules over a commutative ring, with a suitable characteristic polynomial.

Explanations

L1 human

The theorem does not depend on a basis. A linear map satisfies its characteristic polynomial, because each of its matrices does.

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To read this node, read these 21 nodes in this order. The foundations come first, and the list ends with this node.

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 mapdefinitionPolynomial of an endomorphism, definition, no evidence levelPolynomial of an endo…definitionSpanning 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…definitionCayley–Hamilton theorem for linear maps, statement, no evidence levelCayley–Hamilton theor…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. Polynomial of an endomorphism 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. Cayley–Hamilton theorem for linear maps statement

Sources

  1. Web https://en.wikipedia.org/wiki/Cayley–Hamilton_theorem, lead section

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