DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 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.

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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.

Matrix units $E_{i j}$ and the Kronecker delta

Definition

For indices $a, b$ in the same finite index set, the Kronecker delta is

$δ_{a b} : = {\begin{cases} 1, & a = b, \\ 0, & a \neq b . \end{cases}$

Let $i < m$ and $j < n$ . The matrix unit $E_{i j} \in M_{m \times n} (F)$ is the matrix whose $(r, s)$ -entry is

$(E_{i j})_{r s} : = δ_{r i} δ_{s j} .$

Thus $E_{i j}$ has entry $1$ in position $(i, j)$ and entry $0$ everywhere else. When a dimension is zero, no index of that dimension exists and there are no matrix units of the corresponding shape.

Depends on

The vector space $M_{m \times n}(F) := F^{\,m \times n}$ of $m$ by $n$ matrices over a field, with entrywise operations

Used by

dim_F M_m× n(F)=mn and dim_FL(V,W)=(dim_FV)(dim_FW) for finite-dimensional V,W Corollary
BCH truncation fails when higher commutators do not vanish Counterexample
Mₙ(F) as a direct sum of minimal left ideals Example
The affine group of the line Example
The Heisenberg Lie group and algebra Example
Right-invariant fields identify TₑG with the same bracket as left-invariant fields False statement
The exponential map of a Lie group is a group homomorphism False statement
The plus exponential convention is not a homomorphism for left actions False statement
EᵢⱼE_kℓ=δⱼₖE_iℓ Lemma
Similarity is an equivalence relation, and two matrices represent the same endomorphism in two bases exactly when they are similar Theorem

Dependency tree · two levels

6 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

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

Matrix units EijEij​ and the Kronecker delta