Math 2.0

statement theorem

Eigenvalues are the roots of the minimal polynomial

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.

Explanations

L1 human

The minimal polynomial and the characteristic polynomial have the same roots: the eigenvalues. Only the multiplicities of the roots can differ. So the minimal polynomial is a second way to find the eigenvalues.

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To read this node, read these 8 nodes in this order. The foundations come first, and the list ends with this node.

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
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  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

Sources

  1. Web https://en.wikipedia.org/wiki/Minimal_polynomial_(linear_algebra), Properties

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