Minimal polynomial

A minimal polynomial of a linear operator $T:V\to V$ on a finite-dimensional vector space is the unique monic polynomial $m_T(t)\in\mathbb{F}[t]$ of least degree such that

meaning that substituting $T$ into the polynomial yields the zero operator.

The minimal polynomial divides the characteristic polynomial and has the same set of eigenvalues (in a splitting field). It encodes algebraic properties of $T$ more economically than the characteristic polynomial.

Examples:

• For the identity operator $I$, the minimal polynomial is $m_I(t)=t-1$.
• If $T$ is nilpotent and $T^k=0$ with minimal such $k$, then $m_T(t)=t^k$.
• If $T$ is diagonalizable with distinct eigenvalues $\lambda_1,\dots,\lambda_r$ (over a field where they exist), then $m_T(t)=\prod_{i=1}^r (t-\lambda_i)$.