mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.LinearAlgebra.Matrix.Block

Goal state before:
pp: α : Type u_1
m : Type u_3
R : Type v
M : Matrix m m R
b : m → α
inst✝⁴ : CommRing R
inst✝³ : DecidableEq m
inst✝² : Fintype m
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i ↦ b i < k
q : m → Prop := fun i ↦ ¬b i < k
h_sum : M⁻¹.toBlock q p * M.toBlock p p + M⁻¹.toBlock q q * M.toBlock q p = 0
⊢ M.toBlock q p = 0

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Block

Goal state before:
pp: α : Type u_1
m : Type u_3
R : Type v
M : Matrix m m R
b : m → α
inst✝⁴ : CommRing R
inst✝³ : DecidableEq m
inst✝² : Fintype m
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i ↦ b i < k
q : m → Prop := fun i ↦ ¬b i < k
h_sum : M⁻¹.toBlock q p * M.toBlock p p + M⁻¹.toBlock q q * M.toBlock q p = 0
h_zero : M.toBlock q p = 0
⊢ M⁻¹.toBlock q p * M.toBlock p p = 0

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.NonsingularInverse

Goal state before:
pp: m : Type u
n : Type u'
α : Type v
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : CommRing α
inst✝¹ : Fintype m
inst✝ : DecidableEq m
A : Matrix m m α
B : Matrix n n α
hA : IsUnit A.det
hB : ¬IsUnit B.det
h✝ : Nonempty m
⊢ ¬IsUnit (kroneckerMap (fun x1 x2 ↦ x1 * x2) A B).det

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Block

Goal state before:
pp: α : Type u_1
m : Type u_3
R : Type v
M : Matrix m m R
b : m → α
inst✝⁴ : CommRing R
inst✝³ : DecidableEq m
inst✝² : Fintype m
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i ↦ b i < k
q : m → Prop := fun i ↦ ¬b i < k
h_sum : M⁻¹.toBlock q p * M.toBlock p p + M⁻¹.toBlock q q * M.toBlock q p = 0
h_zero : M.toBlock q p = 0
h_mul_eq_zero : M⁻¹.toBlock q p * M.toBlock p p = 0
this : Invertible (M.toBlock p p)
⊢ (fun i ↦ k ≤ b i) = q

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Block

Goal state before:
pp: α : Type u_1
m : Type u_3
R : Type v
M : Matrix m m R
b : m → α
inst✝⁴ : CommRing R
inst✝³ : DecidableEq m
inst✝² : Fintype m
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
⊢ (M⁻¹.toBlock (fun i ↦ k ≤ b i) fun i ↦ b i < k) = 0

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Block

Goal state before:
pp: α : Type u_1
m : Type u_3
R : Type v
M : Matrix m m R
b : m → α
inst✝⁴ : CommRing R
inst✝³ : DecidableEq m
inst✝² : Fintype m
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
this : ∀ (hs : Finset α), image b univ = hs → M⁻¹.BlockTriangular b
⊢ M⁻¹.BlockTriangular b

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Transvection

Goal state before:
pp: 𝕜 : Type u_3
inst✝ : Field 𝕜
r : ℕ
M : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜
⊢ ∃ L L', ((List.map toMatrix L).prod * M * (List.map toMatrix L').prod).IsTwoBlockDiagonal

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.NonsingularInverse

Goal state before:
pp: m : Type u
n : Type u'
α : Type v
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : CommRing α
inst✝¹ : Fintype m
inst✝ : DecidableEq m
A : Matrix m m α
B : Matrix n n α
⊢ (kroneckerMap (fun x1 x2 ↦ x1 * x2) A B)⁻¹ = kroneckerMap (fun x1 x2 ↦ x1 * x2) A⁻¹ B⁻¹

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.NonsingularInverse

Goal state before:
pp: m : Type u
α : Type v
inst✝² : CommRing α
inst✝¹ : Fintype m
inst✝ : DecidableEq m
M : Matrix m m α
h : IsUnit M
N : Matrix m m α
⊢ (M * N * M⁻¹).det = N.det

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.NonsingularInverse

Goal state before:
pp: m : Type u
α : Type v
inst✝² : CommRing α
inst✝¹ : Fintype m
inst✝ : DecidableEq m
M : Matrix m m α
h : IsUnit M
N : Matrix m m α
⊢ (M⁻¹ * N * M).det = N.det

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Block

Goal state before:
pp: α : Type u_1
R : Type v
inst✝⁴ : CommRing R
inst✝³ : LinearOrder α
m : Type u_3
M : Matrix m m R
b : m → α
inst✝² : DecidableEq m
inst✝¹ : Fintype m
inst✝ : Invertible M
hM : M.BlockTriangular b
ih :
  ∀ t ⊂ image b univ,
    ∀ {m : Type u_3} {M : Matrix m m R} {b : m → α} [inst : DecidableEq m] [inst_1 : Fintype m] [Invertible M],
      M.BlockTriangular b → image b univ = t → M⁻¹.BlockTriangular b
i j : m
hij : b j < b i
this : Inhabited m
k : α := (image b univ).max' ⋯
b' : { a // b a < k } → α := fun i ↦ b ↑i
A : Matrix { a // b a < k } { a // b a < k } R := M.toBlock (fun i ↦ b i < k) fun j ↦ b j < k
hbi : b i = k
⊢ M⁻¹.toBlock (fun i ↦ k ≤ b i) (fun i ↦ b i < k) ⟨i, ⋯⟩ ⟨j, ⋯⟩ = 0

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.NonsingularInverse

Goal state before:
pp: m : Type u
α : Type v
inst✝² : CommRing α
inst✝¹ : Fintype m
inst✝ : DecidableEq m
M : Matrix m m α
h : IsUnit M
N : Matrix m m α
⊢ (M * N * M⁻¹).trace = N.trace

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.NonsingularInverse

Goal state before:
pp: m : Type u
α : Type v
inst✝² : CommRing α
inst✝¹ : Fintype m
inst✝ : DecidableEq m
M : Matrix m m α
h : IsUnit M
N : Matrix m m α
⊢ (M⁻¹ * N * M).trace = N.trace

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Transvection

Goal state before:
pp: 𝕜 : Type u_3
inst✝ : Field 𝕜
r : ℕ
IH :
  ∀ (M : Matrix (Fin r) (Fin r) 𝕜),
    ∃ L₀ L₀' D₀, (List.map toMatrix L₀).prod * M * (List.map toMatrix L₀').prod = diagonal D₀
M : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜
L₁ L₁' : List (TransvectionStruct (Fin r ⊕ Unit) 𝕜)
hM : ((List.map toMatrix L₁).prod * M * (List.map toMatrix L₁').prod).IsTwoBlockDiagonal
M' : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜 := (List.map toMatrix L₁).prod * M * (List.map toMatrix L₁').prod
M'' : Matrix (Fin r) (Fin r) 𝕜 := M'.toBlocks₁₁
L₀ L₀' : List (TransvectionStruct (Fin r) 𝕜)
D₀ : Fin r → 𝕜
h₀ : (List.map toMatrix L₀).prod * M'' * (List.map toMatrix L₀').prod = diagonal D₀
c : 𝕜 := M' (inr ()) (inr ())
this :
  (List.map (toMatrix ∘ sumInl Unit) L₀).prod * M' * (List.map (toMatrix ∘ sumInl Unit) L₀').prod =
    diagonal (Sum.elim D₀ fun x ↦ c)
⊢ (List.map toMatrix (List.map (sumInl Unit) L₀ ++ L₁)).prod * M *
      (List.map toMatrix (L₁' ++ List.map (sumInl Unit) L₀')).prod =
    diagonal (Sum.elim D₀ fun x ↦ M' (inr ()) (inr ()))

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.MatrixPolynomialAlgebra

Goal state before:
pp: R : Type u_1
inst✝² : CommSemiring R
n : Type w
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
⊢ matPolyEquiv.symm (C M) = M.map ⇑C

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.MatrixPolynomialAlgebra

Goal state before:
pp: R : Type u_1
inst✝² : CommSemiring R
n : Type w
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
⊢ matPolyEquiv (M.map ⇑C) = C M

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.MatrixPolynomialAlgebra

Goal state before:
pp: R : Type u_1
inst✝² : CommSemiring R
n : Type w
inst✝¹ : DecidableEq n
inst✝ : Fintype n
⊢ matPolyEquiv.symm X = diagonal fun x ↦ X

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.MatrixPolynomialAlgebra

Goal state before:
pp: R : Type u_1
inst✝² : CommSemiring R
n : Type w
inst✝¹ : DecidableEq n
inst✝ : Fintype n
⊢ matPolyEquiv (diagonal fun x ↦ X) = X

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.MatrixPolynomialAlgebra

Goal state before:
pp: R : Type u_1
inst✝² : CommSemiring R
n : Type w
inst✝¹ : DecidableEq n
inst✝ : Fintype n
i j : n
k : ℕ
x : R
⊢ matPolyEquiv (single i j ((monomial k) x)) = (monomial k) (single i j x)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Block

Goal state before:
pp: α : Type u_1
R : Type v
inst✝⁴ : CommRing R
inst✝³ : LinearOrder α
m : Type u_3
M : Matrix m m R
b : m → α
inst✝² : DecidableEq m
inst✝¹ : Fintype m
inst✝ : Invertible M
hM : M.BlockTriangular b
ih :
  ∀ t ⊂ image b univ,
    ∀ {m : Type u_3} {M : Matrix m m R} {b : m → α} [inst : DecidableEq m] [inst_1 : Fintype m] [Invertible M],
      M.BlockTriangular b → image b univ = t → M⁻¹.BlockTriangular b
i j : m
hij : b j < b i
this✝ : Inhabited m
k : α := (image b univ).max' ⋯
b' : { a // b a < k } → α := fun i ↦ b ↑i
A : Matrix { a // b a < k } { a // b a < k } R := M.toBlock (fun i ↦ b i < k) fun j ↦ b j < k
hi : b i < (image b univ).max' ⋯
this : Invertible A
hA : A.BlockTriangular b'
⊢ image b' univ ⊂ image b univ

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Block

Goal state before:
pp: α : Type u_1
R : Type v
inst✝⁴ : CommRing R
inst✝³ : LinearOrder α
m : Type u_3
M : Matrix m m R
b : m → α
inst✝² : DecidableEq m
inst✝¹ : Fintype m
inst✝ : Invertible M
hM : M.BlockTriangular b
ih :
  ∀ t ⊂ image b univ,
    ∀ {m : Type u_3} {M : Matrix m m R} {b : m → α} [inst : DecidableEq m] [inst_1 : Fintype m] [Invertible M],
      M.BlockTriangular b → image b univ = t → M⁻¹.BlockTriangular b
i j : m
hij : b j < b i
this✝ : Inhabited m
k : α := (image b univ).max' ⋯
b' : { a // b a < k } → α := fun i ↦ b ↑i
A : Matrix { a // b a < k } { a // b a < k } R := M.toBlock (fun i ↦ b i < k) fun j ↦ b j < k
hi : b i < (image b univ).max' ⋯
this : Invertible A
hA : A.BlockTriangular b'
hb' : image b' univ ⊂ image b univ
⊢ b' ⟨j, ⋯⟩ < b' ⟨i, hi⟩

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Transvection

Goal state before:
pp: 𝕜 : Type u_3
inst✝ : Field 𝕜
r : ℕ
IH :
  ∀ (M : Matrix (Fin r) (Fin r) 𝕜),
    ∃ L₀ L₀' D₀, (List.map toMatrix L₀).prod * M * (List.map toMatrix L₀').prod = diagonal D₀
M : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜
L₁ L₁' : List (TransvectionStruct (Fin r ⊕ Unit) 𝕜)
hM : ((List.map toMatrix L₁).prod * M * (List.map toMatrix L₁').prod).IsTwoBlockDiagonal
M' : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜 := (List.map toMatrix L₁).prod * M * (List.map toMatrix L₁').prod
M'' : Matrix (Fin r) (Fin r) 𝕜 := M'.toBlocks₁₁
L₀ L₀' : List (TransvectionStruct (Fin r) 𝕜)
D₀ : Fin r → 𝕜
h₀ : (List.map toMatrix L₀).prod * M'' * (List.map toMatrix L₀').prod = diagonal D₀
c : 𝕜 := M' (inr ()) (inr ())
⊢ M' = fromBlocks M'' 0 0 (diagonal fun x ↦ c)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.MatrixPolynomialAlgebra

Goal state before:
pp: R : Type u_1
inst✝² : CommSemiring R
n : Type w
inst✝¹ : DecidableEq n
inst✝ : Fintype n
i j : n
p : R[X]
k : ℕ
⊢ (matPolyEquiv (single i j p)).coeff k = single i j (p.coeff k)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Block

Goal state before:
pp: α : Type u_1
m : Type u_3
R : Type v
M : Matrix m m R
b : m → α
inst✝⁴ : CommRing R
inst✝³ : DecidableEq m
inst✝² : Fintype m
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
⊢ M⁻¹.BlockTriangular b

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Charpoly.Basic

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
n : Type u_4
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
i : n
⊢ M.charmatrix i i = X - C (M i i)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Charpoly.Basic

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
n : Type u_4
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
i j : n
h : i ≠ j
⊢ M.charmatrix i j = -C (M i j)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Charpoly.Basic

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
n : Type u_4
inst✝¹ : DecidableEq n
inst✝ : Fintype n
⊢ charmatrix 0 = (scalar n) X

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Charpoly.Basic

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
n : Type u_4
inst✝¹ : DecidableEq n
inst✝ : Fintype n
d : n → R
⊢ (diagonal d).charmatrix = diagonal fun i ↦ X - C (d i)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Charpoly.Basic

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
n : Type u_4
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
⊢ Mᵀ.charmatrix = M.charmatrixᵀ

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Transvection

Goal state before:
pp: 𝕜 : Type u_3
inst✝ : Field 𝕜
r : ℕ
IH :
  ∀ (M : Matrix (Fin r) (Fin r) 𝕜),
    ∃ L₀ L₀' D₀, (List.map toMatrix L₀).prod * M * (List.map toMatrix L₀').prod = diagonal D₀
M : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜
⊢ ∃ L L' D, (List.map toMatrix L).prod * M * (List.map toMatrix L').prod = diagonal D

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Charpoly.Basic

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
n : Type u_4
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
⊢ matPolyEquiv M.charmatrix = X - C M

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Transvection

Goal state before:
pp: n : Type u_1
p : Type u_2
𝕜 : Type u_3
inst✝⁴ : Field 𝕜
inst✝³ : DecidableEq n
inst✝² : DecidableEq p
inst✝¹ : Fintype n
inst✝ : Fintype p
M : Matrix p p 𝕜
e : p ≃ n
L₀ L₀' : List (TransvectionStruct n 𝕜)
D₀ : n → 𝕜
h₀ : (List.map toMatrix L₀).prod * (reindexAlgEquiv 𝕜 𝕜 e) M * (List.map toMatrix L₀').prod = diagonal D₀
⊢ M = (reindexAlgEquiv 𝕜 𝕜 e.symm) ((reindexAlgEquiv 𝕜 𝕜 e) M)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.MatrixPolynomialAlgebra

Goal state before:
pp: R : Type u_1
inst✝² : CommSemiring R
n : Type w
inst✝¹ : DecidableEq n
inst✝ : Fintype n
m : Matrix n n R[X]
k : ℕ
i j : n
⊢ (matPolyEquiv m).coeff k i j = (m i j).coeff k

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.MatrixPolynomialAlgebra

Goal state before:
pp: R : Type u_1
inst✝² : CommSemiring R
n : Type w
inst✝¹ : DecidableEq n
inst✝ : Fintype n
p : (Matrix n n R)[X]
i j : n
k : ℕ
⊢ p = matPolyEquiv (matPolyEquiv.symm p)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Charpoly.Basic

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
m : Type u_3
n : Type u_4
inst✝³ : DecidableEq m
inst✝² : DecidableEq n
inst✝¹ : Fintype m
inst✝ : Fintype n
M : Matrix n n R
e : n ≃ m
⊢ ((reindex e e) M).charmatrix = (reindex e e) M.charmatrix

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.MatrixPolynomialAlgebra

Goal state before:
pp: R : Type u_1
inst✝² : CommSemiring R
n : Type w
inst✝¹ : DecidableEq n
inst✝ : Fintype n
p : (Matrix n n R)[X]
i j : n
k : ℕ
⊢ (matPolyEquiv.symm p i j).coeff k = p.coeff k i j

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Charpoly.Basic

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing S
n : Type u_4
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
f : R →+* S
⊢ (M.map ⇑f).charmatrix = M.charmatrix.map (Polynomial.map f)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.MatrixPolynomialAlgebra

Goal state before:
pp: R : Type u_1
inst✝² : CommSemiring R
n : Type w
inst✝¹ : DecidableEq n
inst✝ : Fintype n
p : R[X]
⊢ matPolyEquiv (p • 1) = Polynomial.map (algebraMap R (Matrix n n R)) p

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.MatrixPolynomialAlgebra

Goal state before:
pp: R : Type u_1
inst✝² : CommSemiring R
n : Type w
inst✝¹ : DecidableEq n
inst✝ : Fintype n
p : R[X]
M : Matrix n n R[X]
⊢ matPolyEquiv (p • M) = Polynomial.map (algebraMap R (Matrix n n R)) p * matPolyEquiv M

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Charpoly.Basic

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
m : Type u_3
n : Type u_4
inst✝³ : DecidableEq m
inst✝² : DecidableEq n
inst✝¹ : Fintype m
inst✝ : Fintype n
M₁₁ : Matrix m m R
M₁₂ : Matrix m n R
M₂₁ : Matrix n m R
M₂₂ : Matrix n n R
⊢ (fromBlocks M₁₁ M₁₂ M₂₁ M₂₂).charmatrix = fromBlocks M₁₁.charmatrix (-M₁₂.map ⇑C) (-M₂₁.map ⇑C) M₂₂.charmatrix

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Charpoly.Basic

Goal state before:
pp: R : Type u_1
inst✝³ : CommRing R
n : Type u_4
inst✝² : DecidableEq n
inst✝¹ : Fintype n
α : Type u_5
inst✝ : Preorder α
M : Matrix n n R
b : n → α
⊢ M.charmatrix.BlockTriangular b ↔ M.BlockTriangular b

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Transvection

Goal state before:
pp: n : Type u_1
p : Type u_2
𝕜 : Type u_3
inst✝⁴ : Field 𝕜
inst✝³ : DecidableEq n
inst✝² : DecidableEq p
inst✝¹ : Fintype n
inst✝ : Fintype p
M : Matrix p p 𝕜
e : p ≃ n
H : ∃ L L' D, (List.map toMatrix L).prod * (reindexAlgEquiv 𝕜 𝕜 e) M * (List.map toMatrix L').prod = diagonal D
⊢ ∃ L L' D, (List.map toMatrix L).prod * M * (List.map toMatrix L').prod = diagonal D

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.MatrixPolynomialAlgebra

Goal state before:
pp: R : Type u_1
inst✝² : CommSemiring R
n : Type w
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : (Matrix n n R)[X]
r : R
⊢ (matPolyEquiv.symm M).map (eval r) = eval ((scalar n) r) M

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Transvection

Goal state before:
pp: 𝕜 : Type u_3
inst✝² : Field 𝕜
n : Type
inst✝¹ : Fintype n
inst✝ : DecidableEq n
M : Matrix n n 𝕜
this :
  ∀ (cn : ℕ), Fintype.card n = cn → ∃ L L' D, (List.map toMatrix L).prod * M * (List.map toMatrix L').prod = diagonal D
⊢ ∃ L L' D, (List.map toMatrix L).prod * M * (List.map toMatrix L').prod = diagonal D

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.MatrixPolynomialAlgebra

Goal state before:
pp: R : Type u_1
inst✝² : CommSemiring R
n : Type w
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R[X]
r : R
⊢ eval ((scalar n) r) (matPolyEquiv M) = M.map (eval r)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.MatrixPolynomialAlgebra

Goal state before:
pp: R : Type u_1
inst✝² : CommSemiring R
n : Type w
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R[X]
r : R
i j : n
⊢ eval ((scalar n) r) (matPolyEquiv M) i j = eval r (M i j)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Charpoly.Basic

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
m : Type u_3
inst✝¹ : DecidableEq m
inst✝ : Fintype m
M : Matrix m m R
t : R
⊢ eval t M.charpoly = ((scalar m) t - M).det

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Charpoly.Basic

Goal state before:
pp: R : Type u_1
inst✝³ : CommRing R
n : Type u_4
inst✝² : DecidableEq n
inst✝¹ : Fintype n
inst✝ : IsEmpty n
A : Matrix n n R
⊢ A.charpoly = 1

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Charpoly.Basic

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
n : Type u_4
inst✝¹ : DecidableEq n
inst✝ : Fintype n
⊢ charpoly 0 = X ^ Fintype.card n

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Charpoly.Basic

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
n : Type u_4
inst✝¹ : DecidableEq n
inst✝ : Fintype n
d : n → R
⊢ (diagonal d).charpoly = ∏ i, (X - C (d i))

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Charpoly.Basic

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
n : Type u_4
inst✝¹ : DecidableEq n
inst✝ : Fintype n
⊢ charpoly 1 = (X - 1) ^ Fintype.card n

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Charpoly.Basic

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
n : Type u_4
inst✝¹ : DecidableEq n
inst✝ : Fintype n
k : ℕ
⊢ (↑k).charpoly = (X - ↑k) ^ Fintype.card n

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.MatrixPolynomialAlgebra

Goal state before:
pp: R : Type u_1
inst✝² : CommSemiring R
n : Type w
inst✝¹ : DecidableEq n
inst✝ : Fintype n
m : Matrix n n R[X]
i j : n
⊢ (m i j).support ⊆ (matPolyEquiv m).support

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Charpoly.Basic

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
n : Type u_4
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
⊢ Mᵀ.charpoly = M.charpoly

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Charpoly.Basic

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
m : Type u_3
n : Type u_4
inst✝³ : DecidableEq m
inst✝² : DecidableEq n
inst✝¹ : Fintype m
inst✝ : Fintype n
e : n ≃ m
M : Matrix n n R
⊢ ((reindex e e) M).charpoly = M.charpoly

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.MatrixPolynomialAlgebra

Goal state before:
pp: n : Type w
inst✝² : DecidableEq n
inst✝¹ : Fintype n
R : Type u_3
inst✝ : CommRing R
M : Matrix n n R[X]
r : R
⊢ eval r M.det = (eval ((scalar n) r) (matPolyEquiv M)).det

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Charpoly.Basic

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing S
n : Type u_4
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
f : R →+* S
⊢ (M.map ⇑f).charpoly = Polynomial.map f M.charpoly

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Charpoly.Basic

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
m : Type u_3
n : Type u_4
inst✝³ : DecidableEq m
inst✝² : DecidableEq n
inst✝¹ : Fintype m
inst✝ : Fintype n
M₁₁ : Matrix m m R
M₂₁ : Matrix n m R
M₂₂ : Matrix n n R
⊢ (fromBlocks M₁₁ 0 M₂₁ M₂₂).charpoly = M₁₁.charpoly * M₂₂.charpoly

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.MatrixPolynomialAlgebra

Goal state before:
pp: n : Type w
inst✝² : DecidableEq n
inst✝¹ : Fintype n
R : Type u_3
inst✝ : CommRing R
A : Matrix n n R[X]
M : Matrix n n R
⊢ eval 0 (A + X • M.map ⇑C).det = eval 0 A.det

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Charpoly.Basic

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
m : Type u_3
n : Type u_4
inst✝³ : DecidableEq m
inst✝² : DecidableEq n
inst✝¹ : Fintype m
inst✝ : Fintype n
M₁₁ : Matrix m m R
M₁₂ : Matrix m n R
M₂₂ : Matrix n n R
⊢ (fromBlocks M₁₁ M₁₂ 0 M₂₂).charpoly = M₁₁.charpoly * M₂₂.charpoly

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Charpoly.Basic

Goal state before:
pp: R : Type u_1
inst✝³ : CommRing R
n : Type u_4
inst✝² : DecidableEq n
inst✝¹ : Fintype n
M : Matrix n n R
α : Type u_5
inst✝ : DecidableEq α
b : n → α
a : α
⊢ (M.toSquareBlock b a).charmatrix = M.charmatrix.toSquareBlock b a

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Charpoly.Basic

Goal state before:
pp: R : Type u_1
inst✝³ : CommRing R
n : Type u_4
inst✝² : DecidableEq n
inst✝¹ : Fintype n
M : Matrix n n R
α : Type u_5
b : n → α
inst✝ : LinearOrder α
h : M.BlockTriangular b
⊢ M.charpoly = ∏ a ∈ image b univ, (M.toSquareBlock b a).charpoly

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Transvection

Goal state before:
pp: 𝕜 : Type u_3
inst✝² : Field 𝕜
r : ℕ
IH :
  ∀ (n : Type) [inst : Fintype n] [inst_1 : DecidableEq n] (M : Matrix n n 𝕜),
    Fintype.card n = r → ∃ L L' D, (List.map toMatrix L).prod * M * (List.map toMatrix L').prod = diagonal D
n : Type
inst✝¹ : Fintype n
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hn : Fintype.card n = r + 1
⊢ n ≃ Fin r ⊕ Unit

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Charpoly.Basic

Goal state before:
pp: R : Type u_1
inst✝³ : CommRing R
n : Type u_4
inst✝² : DecidableEq n
inst✝¹ : Fintype n
inst✝ : LinearOrder n
M : Matrix n n R
h : M.IsUpperTriangular
⊢ M.charpoly = ∏ i, (X - C (M i i))

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.MatrixPolynomialAlgebra

Goal state before:
pp: R : Type u_1
inst✝³ : CommSemiring R
n : Type w
inst✝² : DecidableEq n
inst✝¹ : Fintype n
S : Type u_3
inst✝ : CommSemiring S
f : S →+* Matrix n n R
⊢ (evalRingHom 0).mapMatrix.comp f.polyToMatrix = f.comp (evalRingHom 0)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Charpoly.Basic

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
n : Type u_4
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
⊢ (aeval M) M.charpoly = 0

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Transvection

Goal state before:
pp: 𝕜 : Type u_3
inst✝² : Field 𝕜
r : ℕ
IH :
  ∀ (n : Type) [inst : Fintype n] [inst_1 : DecidableEq n] (M : Matrix n n 𝕜),
    Fintype.card n = r → ∃ L L' D, (List.map toMatrix L).prod * M * (List.map toMatrix L').prod = diagonal D
n : Type
inst✝¹ : Fintype n
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hn : Fintype.card n = r + 1
e : n ≃ Fin r ⊕ Unit
N : Matrix (Fin r) (Fin r) 𝕜
⊢ Fintype.card (Fin r) = r

Tactic:
by

Goal state after: