Math 2.0

statement theorem

Determinant is the product of the eigenvalues

Statement

Let FF be a field and let AA be an n×nn \times n matrix over FF. Assume that the characteristic polynomial of AA splits over FF as pA(t)=(t−λ1)⋯(t−λn)p_A(t) = (t - \lambda_1) \cdots (t - \lambda_n). Then det⁡A=λ1λ2⋯λn.\det A = \lambda_1 \lambda_2 \cdots \lambda_n.

Remarks

The λi\lambda_i are the eigenvalues of AA, each repeated according to its algebraic multiplicity. The statement follows by comparing the constant coefficients, see the determinant as a coefficient. For example, (2112)\begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix} has eigenvalues 11 and 33, and determinant 33.

Explanations

L1 human

The determinant of a matrix is the product of its eigenvalues, counted with multiplicity. This agrees with the volume picture: in a basis of eigenvectors, the map stretches each direction by its eigenvalue. The volume then changes by the product of these factors.

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To read this node, read these 23 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 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 mapdefinitionDeterminant is a coefficient of the characteristic polynomial, statement, no evidence levelDeterminant is a coef…statementCharacteristic polynomial of a linear map, definition, no evidence levelCharacteristic polyno…definitionAlgebraic multiplicity, definition, no evidence levelAlgebraic multiplicitydefinitionDeterminant is the product of the eigenvalues, statement, no evidence levelDeterminant is the pr…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.
  • 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. 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. Determinant is a coefficient of the characteristic polynomial statement
  21. Characteristic polynomial of a linear map definition
  22. Algebraic multiplicity definition
  23. Determinant is the product of the eigenvalues statement

Sources

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

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