ExampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck 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

$N=E_{01}=\begin{pmatrix}0&1\\0&0\end{pmatrix}\in M_2(F)$

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 $E_{ij}E_{k\ell}=\delta_{jk}E_{i\ell}$ ( $E_{ij}E_{k\ell}=\delta_{jk}E_{i\ell}$ ).

[L2]

Similarity has the form $N=P^{-1}DP$ with $P$ invertible, and every nonzero scalar in a field has an inverse (Similar matrices: $B=P^{-1}AP$ 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 $N\ne0$ . Suppose, for contradiction, that $N=P^{-1}DP$ as in [L2], with $D$ diagonal. Then $0=N^2=P^{-1}D^2P$ , so $D^2=0$ .

assume-contraL1L2

2.1

Every diagonal entry $d$ of $D$ satisfies $d^2=0$ . If $d\ne0$ , the inverse from [L2] gives $d=0$ , a contradiction; hence $D=0$ , and then $N=P^{-1}0P=0$ , 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 · next 3 levels

Direct dependencies and their dependencies through the next three levels: 9 results over 5 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.