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.

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.

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

Dependency tree · 2 levels

Sources

P. E. Leonard, Linear Algebra Notes, §3.4

Matrix units EijEij​ and the Kronecker delta

Definition

Depends on

Dependency tree · 2 levels

Sources

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