PropositionStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-16

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The minimal polynomial of a restriction to an invariant subspace divides the original minimal polynomial

Statement

Let $W$ be a $T$ -invariant subspace of a finite-dimensional vector space. Then the restriction $T ∣_{W} : W \to W$ satisfies

$μ_{T ∣_{W}} ∣ μ_{T} .$

Facts & Assumptions

Given: A finite-dimensional endomorphism $T : V \to V$ and a $T$ -invariant subspace $W \subseteq V$ .

[L1]

A polynomial annihilates an endomorphism exactly when it is divisible by that endomorphism's minimal polynomial (The annihilator ideal is nonzero and has a unique monic generator; $p (T) = 0$ if and only if $μ_{T} ∣ p$ ).

Proof

technique · direct

1.1givenalgebra

Invariance makes $T ∣_{W}$ an endomorphism of $W$ , and for every polynomial $p$ one has $p (T ∣_{W}) = p (T) ∣_{W}$ by induction on powers.

2.1step 1.1L1∎

Since $μ_{T} (T) = 0$ , step 1.1 gives $μ_{T} (T ∣_{W}) = 0$ . Applying [L1] to $T ∣_{W}$ yields $μ_{T ∣_{W}} ∣ μ_{T}$ . This includes $W = 0$ , where $μ_{T ∣_{W}} = 1$ .

Depends on

The annihilator ideal is nonzero and has a unique monic generator; $p(T)=0$ if and only if $\mu_T\mid p$

Used by

The restriction of a diagonalisable endomorphism to an invariant subspace is diagonalisable Theorem

Dependency tree · two levels

7 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, §5, Lemma 5.1 (standard reference, not scraped)