Minimal Polynomial of a Matrix

Minimal vs Characteristic Polynomials

By the Cayley–Hamilton theorem, every n x n matrix A satisfies its degree-n characteristic polynomial p(A) = 0. Therefore, the minimal polynomial m(x) divides p(x) and shares the exact same set of distinct roots (eigenvalues). However, the multiplicity of an eigenvalue in m(x) equals the size of its largest Jordan block, whereas its multiplicity in p(x) equals the sum of sizes of all Jordan blocks for that eigenvalue.

Jordan Block Multiplicity and Coalescence

For a 3x3 matrix in Jordan canonical form with eigenvalues λ and μ, each factor (x - λ)^k in m(x) has exponent k equal to the nilpotency index of (A - λI) restricted to the λ-generalized eigenspace. Consequently, a scalar matrix λI has minimal polynomial x - λ (degree 1), a diagonal matrix diag(λ, μ, μ) with λ ≠ μ has minimal polynomial (x - λ)(x - μ) (degree 2), and a single 3x3 Jordan block has minimal polynomial (x - λ)³ (degree 3).

Worked Example

Calculate the minimal and characteristic polynomials for the diagonal matrix A = diag(1, 2, 2) with λ = 1 and μ = 2:

1. Identify blocks: The diagonal entries are 1, 2, 2, corresponding to three 1x1 Jordan blocks: one at λ = 1 and two at μ = 2.
2. Minimal polynomial: The largest block size for each distinct eigenvalue is 1, so m(x) = (x - 1)(x - 2) = 2 - 3x + x² with ascending coefficients [2, -3, 1] and degree 2.
3. Characteristic polynomial: Including algebraic multiplicities gives p(x) = (x - 1)(x - 2)² = -4 + 8x - 5x² + x³ with ascending coefficients [-4, 8, -5, 1].
4. Verify annihilation: Evaluating m(A) = 2I - 3A + A² gives 2 - 3(1) + 1 = 0 on the first diagonal entry and 2 - 3(2) + 4 = 0 on the second and third diagonal entries, leaving residual 0.

Experiment

Observe how eigenvalue coalescence and superdiagonal 1s change the minimal degree in https://riano.app/module/minimal-polynomial:

1. Select Matrix family Diagonal with Lambda = 1 and Mu = 2, and verify that Minimal degree is 2 with ascending coefficients 2, -3, 1.
2. Change Lambda to 2 so that λ = μ = 2. Watch the minimal degree drop from 2 to 1 with m(x) = x - 2 (ascending coefficients -2, 1), while p(x) becomes -8 + 12x - 6x² + x³.
3. Switch to Jordan 2 plus scalar with Lambda = -2 and Mu = -2 to see that the 2x2 Jordan block forces minimal degree 2 with m(x) = (x + 2)² = 4 + 4x + x².

Limitations

This module focuses on four canonical 3x3 matrix families with integer parameters λ, μ in [-2, 2] so that exact integer arithmetic and block-coalescence transitions can be inspected without floating-point conditioning issues.

Common Pitfalls

Assuming that repeated eigenvalues always imply a non-diagonalizable matrix or a higher-degree minimal polynomial is incorrect: diagonal matrices with repeated eigenvalues still have linear elementary divisors and minimal degree equal to the number of distinct eigenvalues.

References

Algorithms and canonical definitions for minimal and characteristic polynomials of matrices follow standard linear algebra references:

• SageMath Reference Manual: Matrices over General Rings (minpoly and charpoly)