DefinitionDefinition: AI-adaptedProof: 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.

Entrywise ring-matrix operations, rectangular matrix products, identity matrices and transpose

Definition

Let $R$ be a commutative ring. Matrices of one shape are added and scaled entrywise. For $A \in M_{m \times n} (R)$ and $B \in M_{n \times p} (R)$ , their product is $A B \in M_{m \times p} (R)$ with $(A B)_{i k} : = \sum_{j < n} a_{i j} b_{j k} .$ The sum is the finite sum in the additive commutative monoid of $R$ , and is $0$ when $n = 0$ .

The identity matrix $I_{n}$ has entry $1$ on the diagonal and $0$ elsewhere. The transpose $A^{T} \in M_{n \times m} (R)$ is defined by $(A^{T})_{j i} : = a_{i j}$ . The zero matrix has every entry $0$ .

Entrywise ring-matrix operations, rectangular matrix products, identity matrices and transpose

Definition

Depends on

Dependency tree · 2 levels

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