Math 2.0

statement theorem

Trace is the sum 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 tr⁡A=λ1+λ2+⋯+λn.\tr 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 hypothesis always holds over C\C. The statement follows by comparing the coefficients of tn−1t^{n-1}, see the trace as a coefficient. For example, (2112)\begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix} has eigenvalues 11 and 33, and trace 44.

Explanations

L1 human

The trace of a matrix is the sum of its eigenvalues, counted with multiplicity. For a diagonal matrix this is clear, because the eigenvalues are the diagonal entries. The statement says that it stays true for every complex matrix.

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To read this node, read these 24 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 subspacedefinitionTrace of a matrix, definition, no evidence levelTrace of a matrixdefinitionLinear 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…definitionDeterminant, definition, no evidence levelDeterminantdefinitionBasis, definition, no evidence levelBasisdefinitionCharacteristic polynomial, definition, no evidence levelCharacteristic polyno…definitionCoordinate vector, definition, no evidence levelCoordinate vectordefinitionTrace is a coefficient of the characteristic polynomial, statement, no evidence levelTrace is a coefficien…statementMatrix of a linear map, definition, no evidence levelMatrix of a linear mapdefinitionCharacteristic polynomial of a linear map, definition, no evidence levelCharacteristic polyno…definitionAlgebraic multiplicity, definition, no evidence levelAlgebraic multiplicitydefinitionTrace is the sum of the eigenvalues, statement, no evidence levelTrace is the sum of t…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. Trace of a matrix definition
  9. Linear independence definition
  10. Endomorphism definition
  11. Linear span definition
  12. Multilinear map definition
  13. Eigenvalue definition
  14. Spanning set definition
  15. Alternating multilinear map definition
  16. Determinant definition
  17. Basis definition
  18. Characteristic polynomial definition
  19. Coordinate vector definition
  20. Trace is a coefficient of the characteristic polynomial statement
  21. Matrix of a linear map definition
  22. Characteristic polynomial of a linear map definition
  23. Algebraic multiplicity definition
  24. Trace is the sum of the eigenvalues statement

Sources

  1. Web https://en.wikipedia.org/wiki/Trace_(linear_algebra), Trace as the sum of eigenvalues

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