Math 2.0

definition

Characteristic polynomial 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.

Explanations

L1 human

A linear map has a matrix in each basis. All these matrices have the same characteristic polynomial. So we can call it the characteristic polynomial of the map.

What this node needs for W1
  • an inventory row: the row of the inventory that the node accounts for (0 of 1)

The writing levels say how complete the text of a node is. They do not measure its truth.

What this needs

To read this node, read these 19 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 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
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.
  • no evidence level
  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. Spanning set definition
  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

What depends on this

6 nodes use this node, directly or through other nodes.

Sources

  1. Web https://en.wikipedia.org/wiki/Characteristic_polynomial, Formal definition

What links here

Used by