Minimal polynomial

In linear algebra the minimal polynomial of an algebraic object is the monic polynomial of least degree which that object satisfies. Examples include the minimal polynomial of a square matrix, an endomorphism of a vector space or an algebraic number.

The general setting is an algebra A over a field F. We give A the structure of a module over the polynomial ring F[X] by defining the action of $f(x)=\sum _{n=0}^{d}f_{i}X^{i}$ on a to be $f(a)=\sum _{n=0}^{d}f_{i}a^{i}$ where a⁰ is defined to be the unit element of A.

We say that f "annihilates" a, or that a "satisfies" f, if f(a) = 0. The set of polynomials that annihilate a given element a forms an ideal ann(a) in F[X], which is a Euclidean domain. Hence the annihilator ideal is a principal ideal with the minimal polynomial as monic generator.

The minimal polynomial may also be defined as the polynomial of least degree which annihilates a: it then has the property that it divides any other polynomial which annihilates a.

Minimal polynomial of a square matrix

Let A be an n×n matrix over a field F. The powers I=A⁰,A¹,...,A^n² must be linearly dependent since the matrix ring has dimension n² as a vector space over F, and so A satisfies some polynomial. Hence it makes sense to define the minimal polynomial as the monic polynomial of least degree which A satisfies, or which annihilates A.

For an invertible matrix P we have $P^{-1}f(A)P=f(P^{-1)}AP)$. Hence similar matrices have the same minimal polynomial, and we can use this to define the minimal polynomial of an endomorphism as the minimal polynomial of one, and hence any, matric representing it.

Since A satisfies its own characteristic polynomial by the Cayley-Hamilton theorem, we deduce that the minimal polynomial divides the characteristic polynomial. However, the two polynomials have the same set of roots, namely the set of eigenvalues of A.

Minimal polynomial of an endomorphism of a vector space

A similar definition applies to the minimal polynomial of an endomorphism of a finite-dimensional vector space.

Let α be an endomorphism of an n-dimensional vector space over a field F. The powers ι=α⁰,α¹,...,α^n² must be linearly dependent since the endomorphism ring has dimension n² as a vector space over F, and so α satisfies some polynomial. Hence it makes sense to define the minimal polynomial as the monic polynomial of least degree which α satisfies, or which annihilates α.

Minimal polynomial of an algebraic number

The minimal poynomial of an algebraic number α is the rational polynomial of least degree which has α as a root. The degree of the minimal polynomial of α is equal to the degree of the field extension Q(α)/Q.

More generally, if E/F is a field extension and α is in E, then α is said to be algebraic over F if there is a polynomial with coefficients in F which α satsifies. In this case the minimal polynomial of α over F is the polynomial of least degree with coefficients in F which α satisfies. The polynonmial ring F[α] is then a field, and is the simple field extension F(α). This field is a finite dimension al vector space over F, on which multiplication by α is an F-linear endomorphism. The minimal polynomial of α is equal to the minimal polynomial of this endomorphism.