Annihilating polynomial

Definition

Let $F$ be a field, let $V$ be a vector space over $F$, and let $T \colon V \to V$ be an endomorphism. A polynomial $p \in F[x]$ annihilates $T$ when $p(T) = 0$, where $p(T)$ is the value of $p$ at $T$. Such a polynomial is an annihilating polynomial of $T$.

Remarks

The annihilating polynomials of $T$ form an ideal of $F[x]$. If $\dim V = n$ is finite, then a nonzero annihilating polynomial exists, because the $n^2 + 1$ maps $\id_V, T, \dots, T^{n^2}$ are linearly dependent. The monic generator of this ideal is the minimal polynomial of $T$. For a projection $P$, the polynomial $x^2 - x$ annihilates $P$.

Explanations

L1 human

An annihilating polynomial is a polynomial that becomes the zero map when we put $T$ into it. For example, a reflection $R$ satisfies $R^2 = I$, so $x^2 - 1$ annihilates it. Such relations tell us a lot about the possible eigenvalues.