ExampleConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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Computing the minimal polynomial of an explicit $3 \times 3$ idempotent from its annihilating polynomials

Example

For

$A = (\begin{array}{ccc} 0 & - 1 & 1 \\ 1 & 2 & - 1 \\ 1 & 1 & 0 \end{array}),$

one has $A^{2} = A$ , $μ_{A} = x (x - 1)$ , and $χ_{A} = x (x - 1)^{2}$ . Thus $A$ is projection onto its image along its kernel.

Facts & Assumptions

Given: The displayed matrix $A$ .

[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]

Every idempotent endomorphism of a finite-dimensional space is diagonalisable and projects onto its image along its kernel (Every idempotent endomorphism is diagonalisable and is projection onto its image along its kernel).

[L3]

The characteristic polynomial is $\det (x I - A)$ (For $A \in M_{n} (F)$ , the characteristic polynomial is $χ_{A} (x) = \det (x I_{n} - A)$ when $n \geq 1$ , with $χ_{A} (x) = 1$ for the unique $0 \times 0$ matrix).

Verification

technique · direct

1.1

Direct multiplication gives $A^{2} = A$ , while $A \neq 0$ and $A \neq I$ . Thus $x (x - 1)$ annihilates but neither $x$ nor $x - 1$ does; [L1] gives $μ_{A} = x (x - 1)$ .

L1algebra

2.1

Expanding $\det (x I - A)$ gives $x (x - 1)^{2}$ by [L3]. Finally [L2] identifies the image-kernel projection.

L2L3algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 60 results over 16 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, Examples 4.6, 4.9, and 4.16 (standard reference, not scraped)

Computing the minimal polynomial of an explicit 3×33×3 idempotent from its annihilating polynomials