The minimal polynomial of a matrix A is the monic polynomial in A of smallest degree n such that p(A) = sum_(i = 0)^n c_i A^i = 0. The minimal polynomial divides any polynomial q with q(A) = 0 and, in particular, it divides the characteristic polynomial. If the characteristic polynomial factors as char(A)(x) = (x - λ_1)^(n_1) ...(x - λ_k)^(n_k), then its minimal polynomial is given by p(x) = (x - λ_1)^(m_1) ...(x - λ_k)^(m_k) for some positive integers m_i, where the m_i satisfy 1<=m_i<=n_i.

algebraic number minimal polynomial | Cayley-Hamilton theorem | characteristic polynomial | extension field minimal polynomial | rational canonical form