False statementConstruction: AI-adaptedVerification: 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: Every endomorphism has a commuting diagonal-plus-nilpotent decomposition over its base field

Statement

False claim. Every endomorphism $T$ over its base field can be written $T = D + N$ with $D$ diagonalisable, $N$ nilpotent, and $D N = N D$ .

Facts & Assumptions

Given: The real quarter-turn $R = (\begin{array}{cc} 0 & - 1 \\ 1 & 0 \end{array})$ .

[L1]

A commuting family whose characteristic polynomials split is simultaneously triangularisable (A commuting split family is simultaneously triangularisable).

[L2]

A diagonalisable endomorphism has a basis of eigenvectors (A diagonalisable endomorphism is one admitting a basis of eigenvectors, equivalently a diagonal matrix representation).

[L3]

A nilpotent endomorphism has characteristic polynomial $x^{n}$ (Characterisations of a nilpotent endomorphism).

[L4]

An endomorphism is triangularisable exactly when its characteristic polynomial splits ( $T$ is triangularisable iff its minimal polynomial splits iff its characteristic polynomial splits).

Refutation

technique · counterexample

1.1

Suppose $R = D + N$ as claimed. By [L2], $χ_{D}$ splits, while [L3] makes $χ_{N} = x^{2}$ split; commutation and [L1] give one real basis in which both $D$ and $N$ are upper triangular.

assume-contraL1L2L3

2.1

Their sum $R$ is upper triangular in that basis, so [L4] would make $χ_{R}$ split over $R$ .

step 1.1L4algebra

3.1

But direct computation gives $χ_{R} = x^{2} + 1$ , which has no real root. This contradiction refutes the claimed decomposition over the base field.

step 2.1discharge-contradictionalgebra∎

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: 68 results over 20 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

S. Axler, Linear Algebra Done Right, 4th ed., Result 5.43 (standard reference, not scraped)