LemmaStatement: Literature-sourcedProof: AI-adaptedSession-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.

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

Statement

For $E_{ij}\in M_{m\times n}(F)$ and $E_{k\ell}\in M_{n\times p}(F)$ ,

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

Facts & Assumptions

Given: A field $F$ and indices $i<m$ , $j,k<n$ , and $\ell<p$ .

[L1]

The matrix unit $E_{ij}$ has entry $\delta_{ri}\delta_{sj}$ in position $(r,s)$ (Matrix units $E_{ij}$ and the Kronecker delta).

Proof

technique · direct

1.1

For $r<m$ and $s<p$ , the product entry is $(E_{ij}E_{k\ell})_{rs}=\sum_{t<n}\delta_{ri}\delta_{tj}\delta_{tk}\delta_{s\ell}$ .

givenL1

2.1

If $j\ne k$ , every summand is zero. If $j=k$ , only $t=j$ can contribute and the value is $\delta_{ri}\delta_{s\ell}$ .

step 1.1L1

3.1

The two cases combine as $(E_{ij}E_{k\ell})_{rs}=\delta_{jk}(E_{i\ell})_{rs}$ for every entry, proving the matrix identity.

step 2.1L1∎

Depends on

Used by

Mₙ(F) is noncommutative for every n≥2 Corollary
A nonzero square-zero matrix is not similar to any diagonal matrix Example
Two explicit 2 by 2 matrices do not commute Example

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 15 results over 6 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

P. E. Leonard, Linear Algebra Notes, §3.4 (standard reference, not scraped)

EijEkℓ=δjkEiℓE_{ij}E_{k\ell}=\delta_{jk}E_{i\ell}Eij​Ekℓ​=δjk​Eiℓ​