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Full rank coefficient factorizations

Lax342547.RankFactors · concepts/Lax342547/RankFactors.lean · lax-342547

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    Natural Language Statement

    Lemma

    Every binary coefficient matrix factors into two independently injective tuples of exactly its rank. These tuples supply the independent image directions in the nonzero quotient character test.

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    11 concepts
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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on ADescendants are omitted for concepts with more than 10 descendants.
    Evidence

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    1import Lax342547.ProductImages
    2
    3/-!
    4---
    5title: Full rank coefficient factorizations
    6type: lemma
    7---
    8Every binary coefficient matrix factors into two independently injective tuples of exactly its rank. These tuples supply the independent image directions in the nonzero quotient character test.
    9-/
    10
    11namespace Lax342547.RankFactors
    12
    13open Lax342547.MomentSpace
    14
    15axiom independent_rank_factors {I J : Type} [Fintype I] [Fintype J]
    16 (C : Matrix I J Binary) :
    17 ∃ A : Matrix I (Fin C.rank) Binary, ∃ B : Matrix J (Fin C.rank) Binary,
    18 A * B.transpose = C ∧ Function.Injective A.mulVec ∧ Function.Injective B.mulVec
    19
    20end Lax342547.RankFactors
    21
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