Minimal polynomial

In linear algebra the minimal polynomial of a square matrix is the monic polynomial of least degree which the matrix satisfies.

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.

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

The polynomials which annihilate A form an ideal in the ring of polynomials, and this is a principal ideal domain: we deduce that the minimal polynomial actually divides all other polynomials which A satisfies.

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 algebraic number

The minimal poynomial of an algebraic number α is the rational polynomial of least degree which has α as a root.