# Minimal polynomial

In mathematics, the minimal polynomial of an object α is the monic polynomial p of least degree such that p(α)=0. The properties of the minimal polynomial depend on the algebraic structure to which α belongs.

## Field theory

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

In linear algebra, the minimal polynomial of an n-by-n matrix A over a field F is the monic polynomial p(x) 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 minimal polynomial is not always the same as the characteristic polynomial. Consider the matrix ${\displaystyle 4I_{n}}$, which has characteristic polynomial ${\displaystyle (x-4)^{n}}$. However, the minimal polynomial is ${\displaystyle x-4}$, since ${\displaystyle 4I-4I=0}$ as desired, so they are different for ${\displaystyle n\geq 2}$. That the minimal polynomial always divides the characteristic polynomial is a consequence of the Cayley–Hamilton theorem.