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[Merged by Bors] - chore(LinearAlgebra/Matrix/Nondegenerate): generalize lemmas to CommSemiring #41271

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[Merged by Bors] - chore(LinearAlgebra/Matrix/Nondegenerate): generalize lemmas to CommSemiring #41271

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12 changes: 7 additions & 5 deletions Mathlib/LinearAlgebra/Matrix/Nondegenerate.lean
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Original file line number Diff line number Diff line change
Expand Up @@ -46,7 +46,9 @@ structure Nondegenerate (M : Matrix m n R) : Prop where

end Finite

variable {m n R : Type*} [CommRing R] {M : Matrix m n R}
section CommSemiring

variable {m n R : Type*} [CommSemiring R] {M : Matrix m n R}

lemma separatingRight_def [Fintype m] [Fintype n] :
M.SeparatingRight ↔ (∀ w, (∀ v, v ⬝ᵥ M *ᵥ w = 0) → w = 0) := by
Expand All @@ -61,9 +63,7 @@ lemma separatingLeft_def [Fintype m] [Fintype n] :
lemma nondegenerate_def [Fintype m] [Fintype n] :
M.Nondegenerate ↔
(∀ v, (∀ w, v ⬝ᵥ M *ᵥ w = 0) → v = 0) ∧ (∀ w, (∀ v, v ⬝ᵥ M *ᵥ w = 0) → w = 0) := by
constructor
· exact fun h ↦ ⟨separatingLeft_def.mp h.1, separatingRight_def.mp h.2⟩
· exact fun h ↦ ⟨separatingLeft_def.mpr h.1, separatingRight_def.mpr h.2⟩
rw [nondegenerate_iff, separatingLeft_def, separatingRight_def]

theorem separatingLeft_iff_forall_vecMul_eq_zero [Fintype m] [Finite n] :
M.SeparatingLeft ↔ ∀ v, v ᵥ* M = 0 → v = 0 := by
Expand Down Expand Up @@ -140,8 +140,10 @@ theorem Nondegenerate.exists_not_ortho_of_ne_zero' (hM : Nondegenerate M) {w : n
∃ v, v ⬝ᵥ M *ᵥ w ≠ 0 :=
not_forall.mp (mt hM.eq_zero_of_ortho' hw)

end CommSemiring

section Determinant
variable [DecidableEq m] {M : Matrix m m R}
variable {m R : Type*} [CommRing R] [Fintype m] [DecidableEq m] {M : Matrix m m R}

open scoped nonZeroDivisors

Expand Down
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