DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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The annihilator set $Ann (T) = {p \in F [x] : p (T) = 0}$ ; once existence is proved, its unique monic generator $μ_{T}$ is the minimal polynomial

Definition

Let $T : V \to V$ be an endomorphism of a finite-dimensional vector space over $F$ . Its annihilator set is

$Ann (T) : = {p \in F [x] : p (T) = 0},$

where polynomial evaluation at $T$ is that of Polynomial evaluation at an endomorphism: $p (T) = \sum_{k} a_{k} T^{k}$ .

Once The annihilator ideal is nonzero and has a unique monic generator; $p (T) = 0$ if and only if $μ_{T} ∣ p$ ↗ proves that this set is a nonzero principal ideal, the minimal polynomial of $T$ , denoted $μ_{T}$ , is its unique monic generator (The ideal generated by a subset and principal ideals, Degree, leading coefficient and monic polynomial, with the zero polynomial having no degree). Thus existence is not built into the definition. On the zero space, $I_{V} = 0$ , so $1 \in Ann (T)$ and $μ_{T} = 1$ .

Depends on

Used by

Dependency tree · two levels

8 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Keith Conrad, The Minimal Polynomial and Some Applications, §4 (standard reference, not scraped)
Sheldon Axler, Linear Algebra Done Right, 4th ed., §5B (standard reference, not scraped)

The annihilator set Ann⁡(T)={p∈F[x]:p(T)=0}Ann(T)={p∈F[x]:p(T)=0}; once existence is proved, its unique monic generator μTμT​ is the minimal polynomial