Notes on Minimal Polynomial Theorem

Minimal Polynomial Theorem

• Introduction The Minimal Polynomial Theorem addresses properties of the minimal polynomial for a given matrix, denoted as A.

  • Notation:

  • Let A be in Max_n (IF).

Part (a): Existence and Uniqueness of Minimal Polynomial

• The minimal polynomial m_A(x) of A exists and is unique.

  • Uniqueness Reasoning:

  • Linear dependence among A^k for k = 0, 1, …, n-1 implies uniqueness.

  • If a polynomial P(x) satisfies P(A) = 0, then m_A(x) must uniquely define conditions on polynomials of matrix A.

Part (b): Polynomial Division

• If P(x) is any polynomial, then it can be expressed as:

  • P(x) = m_A(x) q(x) for some polynomial q(x).

  • This indicates that m_A(x) divides P(x) evenly.

Part (c): Similarity of Matrices

• Any matrix B similar to A has the same minimal polynomial.

  • Condition for Similarity:

    • Matrix B is similar to A if there exists a matrix P such that B = P A P⁻¹.

    • Both matrices A and B share the same characteristic polynomial:

    • PB(x) = det(B - λI) = det(P A P⁻¹ - λI) = PA(x).

Proof Development - Part (b): Linear Independence

• All powers of A lie in Max_n (IF), leading to a dimensional vector space.

  • By the Maker-Your-Life-Easy Theorem, the set {A^0, A^1, …, A^(n-1)} must be linearly dependent.

  • Question of Independence:

    • If A^0, A^1, …, A^(n-1) are linearly independent, this indicates that k = n.

    • If not, further exploration leads to determining k iteratively down to 1.

Another Formulation of Linear Dependence

• Assume there exist scalars C₀, C₁, …, C_k (not all zero) such that:

  • C₀ A^0 + C₁ A^1 + … + C_k A^k = 0.

  • It follows that C_k cannot be zero due to earlier established conditions of independence.

Definition of Minimal Polynomial

• Proposed minimal polynomial:

  • m(x) = x^k + c{k-1} x^(k-1) + … + c1 x + c_0.

Showing m_A(A) = 0

• Step (i): m(A) = 0 is satisfied.

• Step (ii): The polynomial must be monic, thus the leading coefficient is Cₖ = 1.

• Step (iii): No polynomial of smaller degree satisfies the property P(A) = 0.

  • Show contradiction: Assume such a polynomial exists and generates linear dependence, contradicting uniqueness.

Conclusion: Uniqueness of Minimal Polynomial

• Let m'(x) be another minimal polynomial; show through construction and polynomial division that m_A(x) must equal m'(x).

  • If m(x) and m'(x) share properties under polynomial operation, conclude that both define the same minimal polynomial.

• Minimal polynomial m(x) = x^k + … is concluded as unique.