LemmaStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 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.

Commuting endomorphisms preserve each other's eigenspaces

Statement

If endomorphisms $S, T : V \to V$ commute, then $S (E_{λ} (T)) \subseteq E_{λ} (T)$ for every scalar $λ$ .

Facts & Assumptions

Given: Endomorphisms $S, T$ with $S T = T S$ , a scalar $λ$ , and $v \in E_{λ} (T)$ .

[L1]

The eigenspace is $E_{λ} (T) = {v : T (v) = λ v}$ , including the zero vector (Eigenvalues, eigenvectors, eigenspaces $E_{λ} (T) = \ker (T - λ I)$ , and the spectrum $σ_{F} (T)$ of an endomorphism).

Proof

technique · direct

1.1

Since $T v = λ v$ , commutation gives $T (S v) = S (T v) = S (λ v) = λ S v$ .

givenalgebra

2.1

Therefore $S v \in E_{λ} (T)$ by [L1], whether or not $S v$ is zero.

step 1.1L1∎

Depends on

Eigenvalues, eigenvectors, eigenspaces $E_\lambda(T)=\ker(T-\lambda I)$, and the spectrum $\sigma_F(T)$ of an endomorphism

Used by

A family of diagonalisable endomorphisms of a finite-dimensional space is simultaneously diagonalisable if and only if its members commute pairwise Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 6 results over 5 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Keith Conrad, The Minimal Polynomial and Some Applications, §5 (standard reference, not scraped)