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.

Submatrices and minors of a rectangular matrix

Definition

Let $A = (a_{i j}) \in M_{m \times n} (R)$ be a matrix over a commutative ring (Finite rectangular matrices over a commutative ring, their entries, rows and columns). For increasing lists of distinct row indices $I = (i_{0}, \dots, i_{r - 1})$ and column indices $J = (j_{0}, \dots, j_{s - 1})$ , the submatrix $A [I, J]$ is the $r \times s$ matrix whose $(p, q)$ entry is $a_{i_{p} j_{q}}$ .

When $r = s \geq 1$ , the $(I, J)$ -minor is $\det A [I, J]$ (For $n \geq 1$ , the determinant over a commutative ring by the Leibniz formula, and $∣ \det A ∣$ for a real matrix), and it is called an $r$ -rowed minor. The positive-size condition is part of the terminology here: no determinant of an empty matrix is introduced.

Depends on

Used by

A matrix has rank at least r exactly when it has a nonzero r-rowed minor Lemma

Dependency tree · two levels

13 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

J. M. Lee, Introduction to Smooth Manifolds, Theorem 7.13 (standard reference, not scraped)
L. W. Tu, An Introduction to Manifolds, Section 11.1 (standard reference, not scraped)