Math 2.0

Linear and multilinear algebra; matrix theory Definition

Minimal polynomial

Builds on Annihilating polynomial

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 an endomorphism. The minimal polynomial of TT is the monic polynomial mT∈F[x]m_T \in F[x] of least degree with mT(T)=0m_T(T) = 0. It is the monic generator of the ideal of annihilating polynomials of TT. The minimal polynomial of a square matrix is defined in the same way.

Remarks

The minimal polynomial exists and is unique. It divides each annihilating polynomial, in particular the characteristic polynomial. For I2I_2, the minimal polynomial is x−1x - 1; for (1101)\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}, it is (x−1)2(x - 1)^2. Similar matrices have the same minimal polynomial.

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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 polynomialdefinition
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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

Where this leads

4 pages build on this one.

Further reading

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