Minimal polynomial - Knowlpedia

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

m_T(T)=0,

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

Zero-dimensional convention

On the zero vector space, the minimal polynomial is $1$ : the identity endomorphism is also the zero endomorphism.

Remarks

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 $I$ on a nonzero space, 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 $m_T(t)=\prod_{i=1}^r (t-\lambda_i)$ .