Math 2.0

Linear and multilinear algebra; matrix theory Definition

Characteristic polynomial of a linear map

Builds on Characteristic polynomialEndomorphismMatrix of a linear map

Definition

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 an endomorphism. The characteristic polynomial of TT is pT(t)=p[T]B(t)=det⁡(tIn−[T]B),p_T(t) = p_{[T]_B}(t) = \det(t I_n - [T]_B), the characteristic polynomial of the matrix of TT in a basis BB of VV, where n=dim⁡Vn = \dim V.

Remarks

The polynomial does not depend on BB, see the invariance of the characteristic polynomial. Its degree is dim⁡V\dim V. For the identity map of VV, it is (t−1)n(t - 1)^n.

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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 mapdefinitionSpanning 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…definition
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  13. Alternating multilinear map definition
  14. Basis definition
  15. Determinant definition
  16. Coordinate vector definition
  17. Characteristic polynomial definition
  18. Matrix of a linear map definition
  19. Characteristic polynomial of a linear map definition

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Further reading

  1. Web https://en.wikipedia.org/wiki/Characteristic_polynomial, Formal definition
About this page
Tag0BZ81AJNWF
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Kinddefinition
WritingW0 stub
MSC15A04, 15A18
DefinesCharacteristic polynomial of a linear map
UsesCharacteristic polynomial, Endomorphism, Matrix of a linear map

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