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The minimal polynomial of an n-by-n matrix A over a field F is the polynomial p(x) with leading coefficient 1 over F of least degree such that p(A)=0. Any other polynomial q with q(A) = 0 is a (polynomial) multiple of p.

The following three statements are equivalent:

1. λ∈F is a root of p(x),
2. λ is a root of the characteristic polynomial of A,
3. λ is an eigenvalue of A.

The multiplicity of a root λ of p(x) is the geometric multiplicity of λ and is the size of the largest Jordan block corresponding to λ and the dimension of the corresponding eigenspace.

The minimum polynomial is not always the same as the characteristic polynomial. Consider the matrix ${\displaystyle 4I}$, it has characteristic polynomial ${\displaystyle (x-4)^{4}}$, but ${\displaystyle (x-4)}$ is the minimum polynomial, as ${\displaystyle 4I-4I=0}$ as desired. That the minimal polynomial always divides the characteristic polynomial is a consequence of the Cayley–Hamilton theorem.

In field theory, given a field extension E/F and an element α of E which is algebraic over F, the minimal polynomial of α is the monic polynomial p, with coefficients in F, of least degree such that p(α) = 0. The minimal polynomial is irreducible, and any other non-zero polynomial f with f(α) = 0 is a multiple of p.

• Linear algebra
• Polynomials
• Field theory