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. The powers I=A⁰,A¹,...,A<supn² must be linearly dependent since the matrix ring has dimension n², 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.