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[Merged by Bors] - feat: Matrix.fromRows and Matrix.fromColumns multiplied by vectors #10379

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[Merged by Bors] - feat: Matrix.fromRows and Matrix.fromColumns multiplied by vectors #10379

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6f7f444
feat: Matrix.fromRows_mulVec and Matrix.vecMul_fromColumns
madvorak Feb 9, 2024
e01db91
sorry
madvorak Feb 9, 2024
6b56bd1
prove the lemma
eric-wieser Feb 9, 2024
e452db0
Update Mathlib/Data/Matrix/ColumnRowPartitioned.lean
madvorak Feb 9, 2024
9fc5894
sum_elim_vecMul_fromRows
madvorak Feb 9, 2024
3b618bf
golf
madvorak Feb 9, 2024
3ac06b5
simplify the proof
eric-wieser Feb 9, 2024
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32 changes: 28 additions & 4 deletions Mathlib/Data/Matrix/ColumnRowPartitioned.lean
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Original file line number Diff line number Diff line change
Expand Up @@ -94,11 +94,11 @@ lemma toColumns₂_apply (A : Matrix m (n₁ ⊕ n₂) R) (i : m) (j : n₂) :
(toColumns₂ A) i j = A i (Sum.inr j) := rfl

@[simp]
lemma toColumns₁_fromColumns (A₁ : Matrix m n₁ R) (A₂ : Matrix m n₂ R) :
lemma toColumns₁_fromColumns (A₁ : Matrix m n₁ R) (A₂ : Matrix m n₂ R) :
toColumns₁ (fromColumns A₁ A₂) = A₁ := rfl

@[simp]
lemma toColumns₂_fromColumns (A₁ : Matrix m n₁ R) (A₂ : Matrix m n₂ R) :
lemma toColumns₂_fromColumns (A₁ : Matrix m n₁ R) (A₂ : Matrix m n₂ R) :
toColumns₂ (fromColumns A₁ A₂) = A₂ := rfl

@[simp]
Expand Down Expand Up @@ -144,14 +144,38 @@ section Semiring

variable [Semiring R]

@[simp]
lemma fromRows_mulVec (A₁ : Matrix m₁ n R) (A₂ : Matrix m₂ n R) (v : n → R) :
fromRows A₁ A₂ *ᵥ v = Sum.elim (A₁ *ᵥ v) (A₂ *ᵥ v) := by
ext (_ | _) <;> rfl

@[simp]
lemma vecMul_fromColumns (B₁ : Matrix m n₁ R) (B₂ : Matrix m n₂ R) (v : m → R) :
v ᵥ* fromColumns B₁ B₂ = Sum.elim (v ᵥ* B₁) (v ᵥ* B₂) := by
ext (_ | _) <;> rfl

@[simp]
lemma sum_elim_vecMul_fromRows (B₁ : Matrix m₁ n R) (B₂ : Matrix m₂ n R)
(v₁ : m₁ → R) (v₂ : m₂ → R) :
Sum.elim v₁ v₂ ᵥ* fromRows B₁ B₂ = v₁ ᵥ* B₁ + v₂ ᵥ* B₂ := by
ext
simp [Matrix.vecMul, fromRows, dotProduct]

@[simp]
lemma fromColumns_mulVec_sum_elim (A₁ : Matrix m n₁ R) (A₂ : Matrix m n₂ R)
(v₁ : n₁ → R) (v₂ : n₂ → R) :
fromColumns A₁ A₂ *ᵥ Sum.elim v₁ v₂ = A₁ *ᵥ v₁ + A₂ *ᵥ v₂ := by
ext
simp [Matrix.mulVec, fromColumns]

@[simp]
lemma fromRows_mul (A₁ : Matrix m₁ n R) (A₂ : Matrix m₂ n R) (B : Matrix n m R) :
(fromRows A₁ A₂) * B = fromRows (A₁ * B) (A₂ * B) := by
fromRows A₁ A₂ * B = fromRows (A₁ * B) (A₂ * B) := by
ext (_ | _) _ <;> simp [mul_apply]

@[simp]
lemma mul_fromColumns (A : Matrix m n R) (B₁ : Matrix n n₁ R) (B₂ : Matrix n n₂ R) :
A * (fromColumns B₁ B₂) = fromColumns (A * B₁) (A * B₂) := by
A * fromColumns B₁ B₂ = fromColumns (A * B₁) (A * B₂) := by
ext _ (_ | _) <;> simp [mul_apply]

@[simp]
Expand Down
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