Math 2.0

statement proposition

Minimal polynomial divides every annihilating 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.

Explanations

L1 human

Every polynomial that kills TT is a multiple of the minimal polynomial. If a matrix satisfies A2=AA^2 = A, then its minimal polynomial divides x2−xx^2 - x. So it is xx, x−1x - 1 or x2−xx^2 - x.

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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
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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

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Sources

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

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