ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-08-11

How statement and proof provenance work

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

A nonzero square-zero matrix is not similar to any diagonal matrix

Example

For every field $F$ , the matrix

$N = E_{01} = (\begin{array}{cc} 0 & 1 \\ 0 & 0 \end{array}) \in M_{2} (F)$

is nonzero and satisfies $N^{2} = 0$ , but it is not similar to any diagonal matrix.

Facts & Assumptions

Given: A field $F$ and the displayed matrix $N$ .

[L1]

Matrix units satisfy $E_{i j} E_{k ℓ} = δ_{j k} E_{i ℓ}$ ( $E_{i j} E_{k ℓ} = δ_{j k} E_{i ℓ}$ ).

[L2]

Similarity has the form $N = P^{- 1} D P$ with $P$ invertible, and every nonzero scalar in a field has an inverse (Similar matrices: $B = P^{- 1} A P$ for an invertible $P$ , Field).

Verification

technique · contradiction

1.1

By [L1], $N^{2} = E_{01} E_{01} = 0$ , while the $(0, 1)$ -entry of $N$ is $1$ , so $N \neq 0$ . Suppose, for contradiction, that $N = P^{- 1} D P$ as in [L2], with $D$ diagonal. Then $0 = N^{2} = P^{- 1} D^{2} P$ , so $D^{2} = 0$ .

assume-contraL1L2

2.1

Every diagonal entry $d$ of $D$ satisfies $d^{2} = 0$ . If $d \neq 0$ , the inverse from [L2] gives $d = 0$ , a contradiction; hence $D = 0$ , and then $N = P^{- 1} 0 P = 0$ , contradicting step 1.1. Thus $N$ is not similar to a diagonal matrix.

step 1.1L1L2discharge-contradiction∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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