DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (deepseek-v4-pro + claude-opus-5[1m])audited 2026-08-24

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 equivalence and Smith normal form over a PID

Definition

Let $R$ be a principal ideal domain and let $A, B \in M_{m \times n} (R)$ . The matrices are equivalent when

$B = P A Q$

for some $P \in {GL}_{m} (R)$ and $Q \in {GL}_{n} (R)$ (Invertible square matrices and similarity over a commutative ring).

A Smith normal form of $A$ is an equivalent diagonal rectangular matrix

$diag (d_{1}, \dots, d_{r}, 0, \dots, 0),$

where every $d_{i}$ is nonzero and $d_{1} ∣ d_{2} ∣ \dots ∣ d_{r}$ . Multiplying any $d_{i}$ by a unit gives the same Smith data. The zero matrix has $r = 0$ , and the definition also covers matrices with zero rows or zero columns.

Depends on

Used by

Dependency tree · two levels

11 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

M. Brussel, Finitely Generated Modules over a PID, Definitions 2.0.1 and 2.1.1 (standard reference, not scraped)