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Minimal Polynomial

Jan 06, 2025, 4 min read

  • Let be a linear operator on a vector space . A polynomial is called a minimal polynomial for if is a monic polynomial of least positive degree for which

    Similarly, for matrices this polynomial is the monic polynomial of least degree such that for

  • (Friedberg 7.11) Let be a minimal polynomial for a linear operator on a finite dimensional vector space .

    • If is any polynomial for which , then divides . In particular divides the characteristic polynomial of .
    • is unique.
    • Intuition: This immediately follows from applying the division algorithm.

  • (Friedberg 7.12) Let be a linear operator on a finite-dimensional vector space and let be a basis for . Then the minimal polynomial for is the same as the minimal polynomial for .

  • (Friedberg 7.12.1) Let , the minimal polynomial for is the same as the minimal polynomial for .

  • (Friedberg 7.13) Let be a linear operator on a finite dimensional vector space and let be the minimal polynomial for . A scalar is an eigenvalue of if and only if .

    Hence the characteristic polynomial and the minimal polynomial for have the same zeros.

  • (Friedberg 7.13.1) Let be a linear operator on a finite dimensional vector space with minimal polynomial and characteristic polynomial . Suppose that factors as

    Where are distinct eigenvalues of each with corresponding multiplicities .

  • (Friedberg 7.14) Let be a linear operator on an -dimensional vector space . If is a -cyclic subspace of itself, then the characteristic polynomial and the minimal polynomial for are of the same degree. Hence

  • (Friedberg 7.15) Let be a linear operator on a finite-dimensional vector space . Then is diagonalizable if and only if the minimal polynomial for is of the form

    Where are distinct scalars (eigenvalues)

  • (Friedberg e7.3.12) Let be a linear operator on a finite dimensional vector space and suppose that the characteristic polynomial of splits. Let be the distinct eigenvalues of and for each , let be the order of the largest Jordan block corresponding to in a Jordan canonical form. The minimal polynomial of is

Annihilators

  • Let be a linear operator on a finite dimensional vector space and let . The polynomial is called a -annihilator of if is monic, of least degree, and
  • (Friedberg e7.3.15) Let be a linear operator on a finite dimensional vector space and let .
    • The vector has a unique -annihilator.
    • The -annihilator of divides any polynomial for which
    • If is the -annihilator of and is the -cyclic subspace generated by , then is the minimal polynomial for and
    • The degree of the -annihilator of is if and only if is an eigenvector of .

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