LemmaStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + claude-opus-5[1m])audited 2026-08-24

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A companion block for $(x - λ)^{e}$ is similar to the Jordan block $J_{e} (λ)$

Statement

For $e \geq 1$ and $λ \in F$ , $C ((x - λ)^{e})$ is similar to $J_{e} (λ)$ .

Facts & Assumptions

Given: The power-basis companion matrix of The companion matrix of a monic polynomial and similarity as change of basis (Similarity is an equivalence relation, and two matrices represent the same endomorphism in two bases exactly when they are similar).

[F1]

For $m \geq 1$ and $λ \in F$ , the Jordan block $J_{m} (λ)$ has $λ$ on the diagonal, $1$ on the superdiagonal, and $0$ elsewhere (Jordan blocks, Jordan strings, and their endpoints).

Proof

technique · direct

1.1

In $F [x] / ((x - λ)^{e})$ , the residue classes $1, (x - λ), \dots, (x - λ)^{e - 1}$ form a basis. Reversing their order gives the basis $(x - λ)^{e - 1}, \dots, x - λ, 1$ .

algebra

2.1

Multiplication by $x = λ + (x - λ)$ in the reversed basis has $λ$ on the diagonal and sends each basis vector except the first to itself times $λ$ plus the preceding basis vector. Its matrix therefore has ones on the superdiagonal and is $J_{e} (λ)$ by [F1]. For $e = 1$ it is the matrix $(λ)$ .

step 1.1F1

3.1

In the ordinary power basis $1, x, \dots, x^{e - 1}$ , the same multiplication operator has companion matrix $C ((x - λ)^{e})$ . The two matrices represent one operator in two bases, so they are similar.

step 2.1given∎

Depends on

Used by

Jordan canonical form from the elementary divisors of V_T Theorem

Dependency tree · two levels

14 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

A. Apisa, Wisconsin Math 542, Lemma 37 (standard reference, not scraped)
M. Brussel, Finitely Generated Modules over a PID, Section 7.4 (standard reference, not scraped)

A companion block for (x−λ)e(x−λ)e is similar to the Jordan block Je(λ)Je​(λ)