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Compatibility and codimension of forward/reverse matrix restrictions

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

proven

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    Natural Language Statement

    Lemma

    The overlap calculation in Lemma 5.5: prescribing ZᵀL and LW imposes exactly the common ZᵀLW block of compatibility conditions when Z and W have independent columns. This statement includes zero column ranks.

    Concept map
    1 concept; 4 descendants hidden
    100%
    Proven claimThis conceptRelated conceptA → B: B builds on A
    Evidence

    This concept declares 2 statements. Each proof establishes one of them relative to its assumptions.

    1 compatible_extensions proven

    2 restrictions_codimension proven

    Lean source view on GitHub

    1import Mathlib.LinearAlgebra.Matrix.ToLin
    2import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas
    3import Mathlib.LinearAlgebra.Dimension.Constructions
    4
    5/-!
    6---
    7title: Compatibility and codimension of forward/reverse matrix restrictions
    8type: lemma
    9---
    10The overlap calculation in Lemma 5.5: prescribing ZᵀL and LW imposes
    11exactly the common ZᵀLW block of compatibility conditions when Z and W
    12have independent columns. This statement includes zero column ranks.
    13-/
    14
    15namespace Lax342547.MixerCompatibility
    16
    17variable {K I E F : Type} [Field K] [Fintype I] [Fintype E] [Fintype F]
    18
    19def restrictions (Z : Matrix I E K) (W : Matrix I F K) :
    20 Matrix I I K →ₗ[K] Matrix E I K × Matrix I F K where
    21 toFun L := (Z.transpose * L, L * W)
    22 map_add' L M := by simp [Matrix.mul_add, Matrix.add_mul]
    23 map_smul' c L := by simp [Matrix.mul_smul, Matrix.smul_mul]
    24
    25def compatibility (Z : Matrix I E K) (W : Matrix I F K) :
    26 (Matrix E I K × Matrix I F K) →ₗ[K] Matrix E F K where
    27 toFun p := p.1 * W - Z.transpose * p.2
    28 map_add' x y := by simp [Matrix.add_mul, Matrix.mul_add, add_sub_add_comm]
    29 map_smul' c x := by simp [Matrix.smul_mul, Matrix.mul_smul, smul_sub]
    30
    31axiom compatible_extensions (Z : Matrix I E K) (W : Matrix I F K)
    32 (hZ : Function.Injective Z.mulVec) (hW : Function.Injective W.mulVec)
    33 (A : Matrix E I K) (B : Matrix I F K) :
    34 (∃ L : Matrix I I K, Z.transpose * L = A ∧ L * W = B) ↔ A * W = Z.transpose * B
    35
    36axiom restrictions_codimension (Z : Matrix I E K) (W : Matrix I F K)
    37 (hZ : Function.Injective Z.mulVec) (hW : Function.Injective W.mulVec) :
    38 Module.finrank K (LinearMap.range (restrictions Z W)) + Fintype.card E * Fintype.card F =
    39 Fintype.card E * Fintype.card I + Fintype.card I * Fintype.card F
    40
    41end Lax342547.MixerCompatibility
    42
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