Minimal Polynomial

Statement

Let $K$ be a field and $A$ a square matrix over $K$. The minimal polynomial $\mu_A \in K[x]$ is the unique monic polynomial of smallest degree such that $\mu_A(A) = 0$. It satisfies the divisibility characterisation:

for every polynomial $p \in K[x]$. Equivalently, $\mu_A$ is the generator of the ideal $\{p \in K[x] : p(A) = 0\}$.

Key relationships:

• $\mu_A \mid \chi_A$ (the characteristic polynomial) — from the Cayley–Hamilton TheoremCayley–Hamilton Theorem$$p_A(A) = 0 \;\text{where}\; p_A(\lambda) = \det(\lambda I - A)$$Every square matrix satisfies its own characteristic polynomialRead more →
• $\chi_A \mid \mu_A^n$ where $n = \deg \chi_A$
• $\mu_A$ and $\chi_A$ have the same irreducible factors (same roots, possibly different multiplicities)

Visualization

Compare $A = \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix}$ and $J = \begin{pmatrix} 2 & 1 \\ 0 & 2 \end{pmatrix}$.

For $A$ (scalar matrix):

For $J$ (Jordan block):

Matrix	Char poly $\chi$	Min poly $\mu$	$\mu \mid \chi$?
$A = 2I$	$(x-2)^2$	$x - 2$	Yes: $(x-2)^2 / (x-2)$
$J = J_2(2)$	$(x-2)^2$	$(x-2)^2$	Yes: trivially
$\text{diag}(1,2,3)$	$(x-1)(x-2)(x-3)$	$(x-1)(x-2)(x-3)$	Yes: equal

The minimal polynomial detects the Jordan block structure: $\mu_A$ has $(x - \lambda)^k$ iff the largest Jordan block for $\lambda$ has size $k$.

Proof Sketch

1. Ideal structure. The set $I = \{p \in K[x] : p(A) = 0\}$ is an ideal in $K[x]$. Since $K[x]$ is a PID, $I = (\mu_A)$ for some monic generator $\mu_A$.
2. Cayley–Hamilton. The characteristic polynomial $\chi_A \in I$, so $\mu_A \mid \chi_A$.
3. Divisibility criterion. If $p(A) = 0$ then $p \in I$, so $\mu_A \mid p$.
4. Same roots. If $\lambda$ is an eigenvalue, then $(A - \lambda I)v = 0$ for some $v \ne 0$; evaluating $\mu_A(A)v = 0$ shows $\mu_A(\lambda) = 0$.

Connections

• Cayley–Hamilton TheoremCayley–Hamilton Theorem$$p_A(A) = 0 \;\text{where}\; p_A(\lambda) = \det(\lambda I - A)$$Every square matrix satisfies its own characteristic polynomialRead more → — guarantees $\mu_A \mid \chi_A$; together, $\mu_A$ and $\chi_A$ share the same irreducible factors
• Jordan Canonical FormJordan Canonical Form$$A = P J P^{-1},\quad J = \bigoplus_i J_{n_i}(\lambda_i)$$Every square matrix over an algebraically closed field is similar to a direct sum of Jordan blocks.Read more → — the degree of $(x - \lambda)$ in $\mu_A$ equals the size of the largest Jordan block for eigenvalue $\lambda$
• Diagonalizability CriterionDiagonalizability Criterion$$A \text{ diagonalizable} \iff \mu_A \text{ squarefree}$$A matrix is diagonalizable if and only if its minimal polynomial is squarefree.Read more → — a matrix is diagonalizable if and only if $\mu_A$ is squarefree (no repeated irreducible factor)

import Mathlib.FieldTheory.Minpoly.Field
import Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly

/-- The minimal polynomial divides any annihilating polynomial.
    Over a field, `minpoly.dvd` is the direct Mathlib alias. -/
theorem minpoly_dvd_of_aeval_zero
    {K : Type*} [Field K]
    {n : Type*} [Fintype n] [DecidableEq n]
    (M : Matrix n n K) {p : K[X]}
    (hp : Polynomial.aeval M p = 0) :
    minpoly K M ∣ p :=
  minpoly.dvd K M hp

/-- The minimal polynomial divides the characteristic polynomial.
    Mathlib's `Matrix.minpoly_dvd_charpoly` is the direct alias. -/
theorem minpoly_dvd_charpoly_matrix
    {K : Type*} [Field K]
    {n : Type*} [Fintype n] [DecidableEq n]
    (M : Matrix n n K) :
    minpoly K M ∣ M.charpoly :=
  Matrix.minpoly_dvd_charpoly M