Minimal polynomial

Definition

Let $F$ be a field, let $V$ be a finite-dimensional vector space over $F$, and let $T \colon V \to V$ be an endomorphism. The minimal polynomial of $T$ is the monic polynomial $m_T \in F[x]$ of least degree with $m_T(T) = 0$. It is the monic generator of the ideal of annihilating polynomials of $T$. The minimal polynomial of a square matrix is defined in the same way.

Remarks

The minimal polynomial exists and is unique. It divides each annihilating polynomial, in particular the characteristic polynomial. For $I_2$, the minimal polynomial is $x - 1$; for $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$, it is $(x - 1)^2$. Similar matrices have the same minimal polynomial.

Explanations

L1 human

The minimal polynomial is the shortest polynomial relation that a matrix satisfies. All other relations are multiples of it. Its roots are exactly the eigenvalues of the matrix.