Math 2.0

Linear and multilinear algebra; matrix theory Theorem

Eigenvalues are the roots of the minimal polynomial

Builds on Minimal polynomialEigenvalue

Statement

Let FF be a field, let VV be a finite-dimensional vector space over FF, and let λ∈F\lambda \in F. Let T ⁣:V→VT \colon V \to V be an endomorphism with minimal polynomial mTm_T. Then λ\lambda is an eigenvalue of TT if and only if mT(λ)=0m_T(\lambda) = 0.

Remarks

If T(v)=λvT(v) = \lambda v with v≠0v \ne 0, then 0=mT(T)v=mT(λ)v0 = m_T(T) v = m_T(\lambda) v, so mT(λ)=0m_T(\lambda) = 0. Conversely, if mT=(x−λ)qm_T = (x - \lambda) q, then q(T)≠0q(T) \ne 0 and T−λid⁡VT - \lambda \id_V kills the image of q(T)q(T). So the minimal and the characteristic polynomial have the same roots, possibly with different multiplicities.

Reading path

To understand this page from the ground up, read these 8 pages in order. The foundations come first.

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Vector space, definition, no evidence levelVector spacedefinitionLinear map, definition, no evidence levelLinear mapdefinitionEndomorphism, definition, no evidence levelEndomorphismdefinitionEigenvalue, definition, no evidence levelEigenvaluedefinitionPolynomial of an endomorphism, definition, no evidence levelPolynomial of an endo…definitionAnnihilating polynomial, definition, no evidence levelAnnihilating polynomi…definitionMinimal polynomial, definition, no evidence levelMinimal polynomialdefinitionEigenvalues are the roots of the minimal polynomial, statement, no evidence levelEigenvalues are the r…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Vector space definition
  2. Linear map definition
  3. Endomorphism definition
  4. Eigenvalue definition
  5. Polynomial of an endomorphism definition
  6. Annihilating polynomial definition
  7. Minimal polynomial definition
  8. Eigenvalues are the roots of the minimal polynomial statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Minimal_polynomial_(linear_algebra), Properties
About this page
Tag0QF4ZEFG52
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Kindstatement, theorem
WritingW0 stub
MSC15A18
StatusThis copy of the site has no build data. Run m2 status 0QF4ZEFG52.
UsesMinimal polynomial, Eigenvalue

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