Math 2.0

Linear and multilinear algebra; matrix theory Proposition

Minimal polynomial divides every annihilating polynomial

Builds on Minimal polynomialAnnihilating 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 an endomorphism with minimal polynomial mTm_T. If p∈F[x]p \in F[x] and p(T)=0p(T) = 0, then mTm_T divides pp.

Remarks

The proof divides with remainder: p=qmT+rp = q m_T + r with deg⁡r<deg⁡mT\deg r < \deg m_T. Then r(T)=0r(T) = 0, so r=0r = 0 by the minimality of deg⁡mT\deg m_T.

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Vector space, definition, no evidence levelVector spacedefinitionLinear map, definition, no evidence levelLinear mapdefinitionEndomorphism, definition, no evidence levelEndomorphismdefinitionPolynomial of an endomorphism, definition, no evidence levelPolynomial of an endo…definitionAnnihilating polynomial, definition, no evidence levelAnnihilating polynomi…definitionMinimal polynomial, definition, no evidence levelMinimal polynomialdefinitionMinimal polynomial divides every annihilating polynomial, statement, no evidence levelMinimal polynomial di…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.
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  1. Vector space definition
  2. Linear map definition
  3. Endomorphism definition
  4. Polynomial of an endomorphism definition
  5. Annihilating polynomial definition
  6. Minimal polynomial definition
  7. Minimal polynomial divides every annihilating polynomial statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Minimal_polynomial_(linear_algebra), Formal definition
About this page
Tag0VTAD25A3G
URIm2://0VTAD25A3G
Kindstatement, proposition
WritingW0 stub
MSC15A18
StatusThis copy of the site has no build data. Run m2 status 0VTAD25A3G.
UsesMinimal polynomial, Annihilating polynomial

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