Minimal Polynomial

Let $T$ be a linear operator on a vector space $V$ over a field $F$. The minimal polynomial of $T$ is the unique monic polynomial $m_T(t)\in F[t]$ (in the polynomial ring over $F$) of least degree such that

where polynomial evaluation is defined by: if $p(t)=a_0+a_1t+\cdots+a_dt^d$, then

(with $T^d$ denoting $d$-fold composition and $I$ the identity operator on $V$).

In finite dimensions, $m_T$ always exists and divides the characteristic polynomial $\chi_T$. Existence is guaranteed, for instance, by the Cayley-Hamilton theorem (which provides a nonzero polynomial that annihilates $T$).

Examples:

• If $T=I$ on $V$, then $m_T(t)=t-1$.
• If $T$ is nilpotent with $T^k=0$ and $k$ minimal, then $m_T(t)=t^k$.
• If $T$ is diagonalizable with distinct eigenvalues $\lambda_1,\dots,\lambda_r$ (over a splitting field), then $m_T(t)=\prod_{i=1}^r (t-\lambda_i)$.