False statementConstruction: 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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FALSE: Equal characteristic and minimal polynomials imply similarity

Statement

False claim. Two matrices with the same characteristic polynomial and the same minimal polynomial must be similar.

Facts & Assumptions

Given: For any $λ \in F$ , $A = J_{2} (λ) \oplus J_{2} (λ), B = J_{2} (λ) \oplus J_{1} (λ) \oplus J_{1} (λ) .$

[L1]

Jordan block sizes give the characteristic polynomial by total size and the minimal polynomial by largest size (For split operators, Jordan blocks read off eigenspace multiplicities and both canonical polynomials).

[L2]

Split matrices are similar exactly when their Jordan block multisets agree (Split matrices are similar exactly when their Jordan block multisets agree).

Refutation

technique · counterexample

1.1

Both matrices have total size four and largest block size two, so [L1] gives $χ_{A} = χ_{B} = (x - λ)^{4}$ and $μ_{A} = μ_{B} = (x - λ)^{2}$ .

L1algebra

1.2

Nevertheless $rank (A - λ I) = 2$ while $rank (B - λ I) = 1$ , and their block multisets are respectively ${2, 2}$ and ${2, 1, 1}$ .

algebra

2.1

Fact [L2] therefore says that $A$ and $B$ are not similar, refuting the claim.

step 1.2L2∎

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: 30 results over 7 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

K. Hoffman and R. Kunze, Linear Algebra, 2nd ed., Section 6.8, Exercise 9 (standard reference, not scraped)