ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck 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.

Minimal polynomials of scalar and diagonal endomorphisms, including the zero-dimensional case

Example

The zero-dimensional endomorphism has minimal polynomial $1$ . On a nonzero space, the scalar endomorphism $λ I$ has minimal polynomial $x - λ$ . A diagonal matrix whose distinct diagonal values are $λ_{0}, \dots, λ_{r - 1}$ has minimal polynomial $\prod_{i < r} (x - λ_{i})$ , regardless of repetitions on the diagonal.

Facts & Assumptions

Given: The endomorphisms described in the Example.

[L1]

An annihilating polynomial is exactly a multiple of the minimal polynomial (The annihilator ideal is nonzero and has a unique monic generator; $p (T) = 0$ if and only if $μ_{T} ∣ p$ ).

[L2]

A diagonalisable endomorphism has a minimal polynomial that is a product of distinct linear factors (An endomorphism is diagonalisable if and only if its minimal polynomial is a product of distinct linear factors).

Verification

technique · direct

1.1L1algebra

On the zero space, $I = 0$ , so $1$ annihilates and [L1] gives $μ = 1$ . On a nonzero space, $(λ I - λ I) = 0$ , while no nonzero constant polynomial annihilates; hence $μ_{λ I} = x - λ$ .

2.1L1L2algebra∎

For the diagonal matrix, a polynomial vanishes on the matrix exactly when it vanishes at every displayed diagonal value. The monic polynomial of least degree with those distinct roots is $\prod_{i} (x - λ_{i})$ , agreeing with [L2].

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 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

Sheldon Axler, Linear Algebra Done Right, 4th ed., §5B (standard reference, not scraped)
Anthony W. Knapp, Basic Algebra, 2nd ed., Ch. V, §3 (standard reference, not scraped)