Math 2.0

definition

Algebraic multiplicity

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 a linear map. Let λ\lambda be an eigenvalue of TT. The algebraic multiplicity of λ\lambda is the largest integer m≥1m \ge 1 such that (t−λ)m(t - \lambda)^m divides the characteristic polynomial pT(t)p_T(t).

Remarks

The sum of the algebraic multiplicities of all eigenvalues is at most dim⁡V\dim V, with equality when pTp_T splits over FF. For (1101)\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}, the eigenvalue 11 has algebraic multiplicity 22. The algebraic multiplicity is at least the geometric multiplicity.

Explanations

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An eigenvalue can be a root of the characteristic polynomial several times. The algebraic multiplicity counts how many times. For the identity matrix of size nn, the eigenvalue 11 has algebraic multiplicity nn.

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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…definitionAlgebraic multiplicity, definition, no evidence levelAlgebraic multiplicitydefinition
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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. Algebraic multiplicity definition

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Sources

  1. Web https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors, Algebraic multiplicity

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