False statementConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-24

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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.

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FALSE: characteristic and minimal polynomials determine similarity

Statement

False claim. Two square matrices over a field having the same characteristic polynomial and the same minimal polynomial are similar.

Facts & Assumptions

Given: The minimal and characteristic polynomial dictionaries of On a nonzero finite-dimensional space, the largest invariant factor is the minimal polynomial and The product of the invariant factors is the characteristic polynomial.

[L1]

Two matrices are similar exactly when their invariant factors agree (Two matrices are similar exactly when their invariant factors agree).

Refutation

technique · direct

1.1

Let $A = J_{2} (0) \oplus J_{2} (0)$ and $B = J_{2} (0) \oplus J_{1} (0) \oplus J_{1} (0)$ . Both are four-dimensional nilpotent matrices with characteristic polynomial $x^{4}$ and largest block size two, hence minimal polynomial $x^{2}$ .

algebra

2.1

The invariant factors of $A$ are $x^{2}, x^{2}$ , while those of $B$ are $x, x, x^{2}$ ; equivalently, their ranks are $2$ and $1$ . By [L1] they are not similar, despite the equal characteristic and minimal polynomials.

step 1.1L1∎

Remarks

This witness agrees with FALSE: Equal characteristic and minimal polynomials imply similarity, but the proof here is self-contained because that published item lies on an examples page and is not a dependency.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

10 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, Lecture 12 worksheet, Problem 3 (standard reference, not scraped)