Math 2.0

statement theorem

Eigenvalues are the roots of the characteristic polynomial

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. Let pTp_T be the characteristic polynomial of TT, and let λ∈F\lambda \in F. Then λ\lambda is an eigenvalue of TT if and only if pT(λ)=0p_T(\lambda) = 0.

Remarks

The proof: λ\lambda is an eigenvalue if and only if λid⁡V−T\lambda \id_V - T is not injective, if and only if det⁡(λid⁡V−T)=0\det(\lambda \id_V - T) = 0. So TT has at most dim⁡V\dim V eigenvalues. For A=(2112)A = \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}, pA(t)=t2−4t+3=(t−1)(t−3)p_A(t) = t^2 - 4t + 3 = (t - 1)(t - 3), so the eigenvalues are 11 and 33.

Explanations

L1 human

To find the eigenvalues of a matrix, compute its characteristic polynomial and find its roots. Each root is an eigenvalue, and each eigenvalue is a root. Then, for each eigenvalue λ\lambda, solve (A−λI)x=0(A - \lambda I)x = 0 to find the eigenvectors.

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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 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 mapdefinitionCharacteristic polynomial of a linear map, definition, no evidence levelCharacteristic polyno…definitionEigenvalues are the roots of the characteristic polynomial, statement, no evidence levelEigenvalues are the r…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. 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. Characteristic polynomial of a linear map definition
  21. Eigenvalues are the roots of the characteristic polynomial statement

Sources

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

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