Nonsingular inverses #

THIS FILE IS SYNCHRONIZED WITH MATHLIB4. Any changes to this file require a corresponding PR to mathlib4.

In this file, we define an inverse for square matrices of invertible determinant.

For matrices that are not square or not of full rank, there is a more general notion of pseudoinverses which we do not consider here.

The definition of inverse used in this file is the adjugate divided by the determinant. We show that dividing the adjugate by det A (if possible), giving a matrix A⁻¹ (nonsing_inv), will result in a multiplicative inverse to A.

Note that there are at least three different inverses in mathlib:

A⁻¹ (has_inv.inv): alone, this satisfies no properties, although it is usually used in conjunction with group or group_with_zero. On matrices, this is defined to be zero when no inverse exists.
⅟A (inv_of): this is only available in the presence of [invertible A], which guarantees an inverse exists.
ring.inverse A: this is defined on any monoid_with_zero, and just like ⁻¹ on matrices, is defined to be zero when no inverse exists.

We start by working with invertible, and show the main results:

After this we define matrix.has_inv and show it matches ⅟A and ring.inverse A. The rest of the results in the file are then about A⁻¹

References #

https://en.wikipedia.org/wiki/Cramer's_rule#Finding_inverse_matrix

Tags #

matrix inverse, cramer, cramer's rule, adjugate

Matrices are `invertible` iff their determinants are #

source

def matrix.invertible_of_det_invertible {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] (A : matrix n n α) [invertible A.det] :

invertible A

If A.det has a constructive inverse, produce one for A.

Equations

A.invertible_of_det_invertible = {inv_of := ⅟ A.det • A.adjugate, inv_of_mul_self := _, mul_inv_of_self := _}

source

theorem matrix.inv_of_eq {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] (A : matrix n n α) [invertible A.det] [invertible A] :

⅟ A = ⅟ A.det • A.adjugate

source

def matrix.det_invertible_of_left_inverse {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] (A B : matrix n n α) (h : B.mul A = 1) :

invertible A.det

A.det is invertible if A has a left inverse.

Equations

A.det_invertible_of_left_inverse B h = {inv_of := B.det, inv_of_mul_self := _, mul_inv_of_self := _}

source

def matrix.det_invertible_of_right_inverse {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] (A B : matrix n n α) (h : A.mul B = 1) :

invertible A.det

A.det is invertible if A has a right inverse.

Equations

A.det_invertible_of_right_inverse B h = {inv_of := B.det, inv_of_mul_self := _, mul_inv_of_self := _}

source

def matrix.det_invertible_of_invertible {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] (A : matrix n n α) [invertible A] :

invertible A.det

If A has a constructive inverse, produce one for A.det.

Equations

A.det_invertible_of_invertible = A.det_invertible_of_left_inverse (⅟ A) _

source

theorem matrix.det_inv_of {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] (A : matrix n n α) [invertible A] [invertible A.det] :

(⅟ A).det = ⅟ A.det

source

@[simp]

theorem matrix.invertible_equiv_det_invertible_apply {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] (A : matrix n n α) [invertible A] :

⇑(A.invertible_equiv_det_invertible) _inst_4 = A.det_invertible_of_invertible

source

@[simp]

theorem matrix.invertible_equiv_det_invertible_symm_apply {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] (A : matrix n n α) [invertible A.det] :

⇑(A.invertible_equiv_det_invertible.symm) _inst_4 = A.invertible_of_det_invertible

source

def matrix.invertible_equiv_det_invertible {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] (A : matrix n n α) :

invertible A ≃ invertible A.det

Together matrix.det_invertible_of_invertible and matrix.invertible_of_det_invertible form an equivalence, although both sides of the equiv are subsingleton anyway.

Equations

A.invertible_equiv_det_invertible = {to_fun := A.det_invertible_of_invertible, inv_fun := A.invertible_of_det_invertible, left_inv := _, right_inv := _}

source

theorem matrix.mul_eq_one_comm {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] {A B : matrix n n α} :

A.mul B = 1 ↔ B.mul A = 1

source

def matrix.invertible_of_left_inverse {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] (A B : matrix n n α) (h : B.mul A = 1) :

invertible A

We can construct an instance of invertible A if A has a left inverse.

Equations

A.invertible_of_left_inverse B h = {inv_of := B, inv_of_mul_self := h, mul_inv_of_self := _}

source

def matrix.invertible_of_right_inverse {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] (A B : matrix n n α) (h : A.mul B = 1) :

invertible A

We can construct an instance of invertible A if A has a right inverse.

Equations

A.invertible_of_right_inverse B h = {inv_of := B, inv_of_mul_self := _, mul_inv_of_self := h}

source

@[protected, instance]

def matrix.invertible_transpose {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] (A : matrix n n α) [invertible A] :

invertible A.transpose

The transpose of an invertible matrix is invertible.

Equations

A.invertible_transpose = A.transpose.invertible_of_det_invertible

source

def matrix.invertible__of_invertible_transpose {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] (A : matrix n n α) [invertible A.transpose] :

invertible A

A matrix is invertible if the transpose is invertible.

Equations

A.invertible__of_invertible_transpose = _.mpr A.transpose.invertible_transpose

source

def matrix.invertible_of_invertible_conj_transpose {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] (A : matrix n n α) [star_ring α] [invertible A.conj_transpose] :

invertible A

A matrix is invertible if the conjugate transpose is invertible.

Equations

A.invertible_of_invertible_conj_transpose = _.mpr (invertible.star A.conj_transpose)

source

def matrix.unit_of_det_invertible {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] (A : matrix n n α) [invertible A.det] :

(matrix n n α)ˣ

Given a proof that A.det has a constructive inverse, lift A to (matrix n n α)ˣ

Equations

A.unit_of_det_invertible = unit_of_invertible A

source

theorem matrix.is_unit_iff_is_unit_det {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] (A : matrix n n α) :

is_unit A ↔ is_unit A.det

When lowered to a prop, matrix.invertible_equiv_det_invertible forms an iff.

Variants of the statements above with `is_unit` #

source

theorem matrix.is_unit_det_of_invertible {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] (A : matrix n n α) [invertible A] :

is_unit A.det

source

theorem matrix.is_unit_of_left_inverse {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] {A B : matrix n n α} (h : B.mul A = 1) :

is_unit A

source

theorem matrix.is_unit_of_right_inverse {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] {A B : matrix n n α} (h : A.mul B = 1) :

is_unit A

source

theorem matrix.is_unit_det_of_left_inverse {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] {A B : matrix n n α} (h : B.mul A = 1) :

is_unit A.det

source

theorem matrix.is_unit_det_of_right_inverse {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] {A B : matrix n n α} (h : A.mul B = 1) :

is_unit A.det

source

theorem matrix.det_ne_zero_of_left_inverse {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] {A B : matrix n n α} [nontrivial α] (h : B.mul A = 1) :

A.det ≠ 0

source

theorem matrix.det_ne_zero_of_right_inverse {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] {A B : matrix n n α} [nontrivial α] (h : A.mul B = 1) :

A.det ≠ 0

source

theorem matrix.is_unit_det_transpose {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] (A : matrix n n α) (h : is_unit A.det) :

is_unit A.transpose.det

A noncomputable `has_inv` instance #

source

@[protected, instance]

noncomputable def matrix.has_inv {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] :

has_inv (matrix n n α)

The inverse of a square matrix, when it is invertible (and zero otherwise).

Equations

matrix.has_inv = {inv := λ (A : matrix n n α), ring.inverse A.det • A.adjugate}

source

theorem matrix.inv_def {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] (A : matrix n n α) :

A⁻¹ = ring.inverse A.det • A.adjugate

source

theorem matrix.nonsing_inv_apply_not_is_unit {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] (A : matrix n n α) (h : ¬is_unit A.det) :

A⁻¹ = 0

source

theorem matrix.nonsing_inv_apply {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] (A : matrix n n α) (h : is_unit A.det) :

A⁻¹ = ↑(h.unit)⁻¹ • A.adjugate

source

@[simp]

theorem matrix.inv_of_eq_nonsing_inv {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] (A : matrix n n α) [invertible A] :

⅟ A = A⁻¹

The nonsingular inverse is the same as inv_of when A is invertible.

source

@[simp, norm_cast]

theorem matrix.coe_units_inv {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] (A : (matrix n n α)ˣ) :

↑A⁻¹ = (↑A)⁻¹

Coercing the result of units.has_inv is the same as coercing first and applying the nonsingular inverse.

source

theorem matrix.nonsing_inv_eq_ring_inverse {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] (A : matrix n n α) :

A⁻¹ = ring.inverse A

The nonsingular inverse is the same as the general ring.inverse.

source

theorem matrix.transpose_nonsing_inv {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] (A : matrix n n α) :

A⁻¹.transpose = (A.transpose)⁻¹

source

theorem matrix.conj_transpose_nonsing_inv {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] (A : matrix n n α) [star_ring α] :

A⁻¹.conj_transpose = (A.conj_transpose)⁻¹

source

@[simp]

theorem matrix.mul_nonsing_inv {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] (A : matrix n n α) (h : is_unit A.det) :

A.mul A⁻¹ = 1

The nonsing_inv of A is a right inverse.

source

@[simp]

theorem matrix.nonsing_inv_mul {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] (A : matrix n n α) (h : is_unit A.det) :

A⁻¹.mul A = 1

The nonsing_inv of A is a left inverse.

source

@[protected, instance]

def matrix.has_inv.inv.invertible {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] (A : matrix n n α) [invertible A] :

invertible A⁻¹

Equations

matrix.has_inv.inv.invertible A = _.mpr invertible_inv_of

source

@[simp]

theorem matrix.inv_inv_of_invertible {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] (A : matrix n n α) [invertible A] :

A⁻¹ ⁻¹ = A

source

@[simp]

theorem matrix.mul_nonsing_inv_cancel_right {m : Type u} {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] (A : matrix n n α) (B : matrix m n α) (h : is_unit A.det) :

(B.mul A).mul A⁻¹ = B

source

@[simp]

theorem matrix.mul_nonsing_inv_cancel_left {m : Type u} {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] (A : matrix n n α) (B : matrix n m α) (h : is_unit A.det) :

A.mul (A⁻¹.mul B) = B

source

@[simp]

theorem matrix.nonsing_inv_mul_cancel_right {m : Type u} {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] (A : matrix n n α) (B : matrix m n α) (h : is_unit A.det) :

(B.mul A⁻¹).mul A = B

source

@[simp]

theorem matrix.nonsing_inv_mul_cancel_left {m : Type u} {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] (A : matrix n n α) (B : matrix n m α) (h : is_unit A.det) :

A⁻¹.mul (A.mul B) = B

source

@[simp]

theorem matrix.mul_inv_of_invertible {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] (A : matrix n n α) [invertible A] :

A.mul A⁻¹ = 1

source

@[simp]

theorem matrix.inv_mul_of_invertible {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] (A : matrix n n α) [invertible A] :

A⁻¹.mul A = 1

source

@[simp]

theorem matrix.mul_inv_cancel_right_of_invertible {m : Type u} {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] (A : matrix n n α) (B : matrix m n α) [invertible A] :

(B.mul A).mul A⁻¹ = B

source

@[simp]

theorem matrix.mul_inv_cancel_left_of_invertible {m : Type u} {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] (A : matrix n n α) (B : matrix n m α) [invertible A] :

A.mul (A⁻¹.mul B) = B

source

@[simp]

theorem matrix.inv_mul_cancel_right_of_invertible {m : Type u} {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] (A : matrix n n α) (B : matrix m n α) [invertible A] :

(B.mul A⁻¹).mul A = B

source

@[simp]

theorem matrix.inv_mul_cancel_left_of_invertible {m : Type u} {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] (A : matrix n n α) (B : matrix n m α) [invertible A] :

A⁻¹.mul (A.mul B) = B

source

theorem matrix.inv_mul_eq_iff_eq_mul_of_invertible {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] (A B C : matrix n n α) [invertible A] :

A⁻¹.mul B = C ↔ B = A.mul C

source

theorem matrix.mul_inv_eq_iff_eq_mul_of_invertible {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] (A B C : matrix n n α) [invertible A] :

B.mul A⁻¹ = C ↔ B = C.mul A

source

theorem matrix.nonsing_inv_cancel_or_zero {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] (A : matrix n n α) :

A⁻¹.mul A = 1 ∧ A.mul A⁻¹ = 1 ∨ A⁻¹ = 0

source

theorem matrix.det_nonsing_inv_mul_det {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] (A : matrix n n α) (h : is_unit A.det) :

A⁻¹.det * A.det = 1

source

@[simp]

theorem matrix.det_nonsing_inv {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] (A : matrix n n α) :

A⁻¹.det = ring.inverse A.det

source

theorem matrix.is_unit_nonsing_inv_det {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] (A : matrix n n α) (h : is_unit A.det) :

is_unit A⁻¹.det

source

@[simp]

theorem matrix.nonsing_inv_nonsing_inv {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] (A : matrix n n α) (h : is_unit A.det) :

A⁻¹ ⁻¹ = A

source

theorem matrix.is_unit_nonsing_inv_det_iff {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] {A : matrix n n α} :

is_unit A⁻¹.det ↔ is_unit A.det

source

noncomputable def matrix.invertible_of_is_unit_det {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] (A : matrix n n α) (h : is_unit A.det) :

invertible A

A version of matrix.invertible_of_det_invertible with the inverse defeq to A⁻¹ that is therefore noncomputable.

Equations

A.invertible_of_is_unit_det h = {inv_of := A⁻¹, inv_of_mul_self := _, mul_inv_of_self := _}

source

noncomputable def matrix.nonsing_inv_unit {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] (A : matrix n n α) (h : is_unit A.det) :

(matrix n n α)ˣ

A version of matrix.units_of_det_invertible with the inverse defeq to A⁻¹ that is therefore noncomputable.

Equations

A.nonsing_inv_unit h = unit_of_invertible A

source

theorem matrix.unit_of_det_invertible_eq_nonsing_inv_unit {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] (A : matrix n n α) [invertible A.det] :

A.unit_of_det_invertible = A.nonsing_inv_unit _

source

theorem matrix.inv_eq_left_inv {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] {A B : matrix n n α} (h : B.mul A = 1) :

A⁻¹ = B

If matrix A is left invertible, then its inverse equals its left inverse.

source

theorem matrix.inv_eq_right_inv {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] {A B : matrix n n α} (h : A.mul B = 1) :

A⁻¹ = B

If matrix A is right invertible, then its inverse equals its right inverse.

source

theorem matrix.left_inv_eq_left_inv {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] {A B C : matrix n n α} (h : B.mul A = 1) (g : C.mul A = 1) :

B = C

The left inverse of matrix A is unique when existing.

source

theorem matrix.right_inv_eq_right_inv {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] {A B C : matrix n n α} (h : A.mul B = 1) (g : A.mul C = 1) :

B = C

The right inverse of matrix A is unique when existing.

source

theorem matrix.right_inv_eq_left_inv {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] {A B C : matrix n n α} (h : A.mul B = 1) (g : C.mul A = 1) :

B = C

The right inverse of matrix A equals the left inverse of A when they exist.

source

theorem matrix.inv_inj {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] {A B : matrix n n α} (h : A⁻¹ = B⁻¹) (h' : is_unit A.det) :

A = B

source

@[simp]

theorem matrix.inv_zero {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] :

0⁻¹ = 0

source

@[protected, instance]

noncomputable def matrix.inv_one_class {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] :

inv_one_class (matrix n n α)

Equations

matrix.inv_one_class = {one := 1, inv := has_inv.inv matrix.has_inv, inv_one := _}

source

theorem matrix.inv_smul {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] (A : matrix n n α) (k : α) [invertible k] (h : is_unit A.det) :

(k • A)⁻¹ = ⅟ k • A⁻¹

source

theorem matrix.inv_smul' {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] (A : matrix n n α) (k : αˣ) (h : is_unit A.det) :

(k • A)⁻¹ = k⁻¹ • A⁻¹

source

theorem matrix.inv_adjugate {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] (A : matrix n n α) (h : is_unit A.det) :

(A.adjugate)⁻¹ = (h.unit)⁻¹ • A

source

def matrix.diagonal_invertible {n : Type u'} [fintype n] [decidable_eq n] {α : Type u_1} [non_assoc_semiring α] (v : n → α) [invertible v] :

invertible (matrix.diagonal v)

diagonal v is invertible if v is

Equations

matrix.diagonal_invertible v = invertible.map (matrix.diagonal_ring_hom n α) v

source

theorem matrix.inv_of_diagonal_eq {n : Type u'} [fintype n] [decidable_eq n] {α : Type u_1} [semiring α] (v : n → α) [invertible v] [invertible (matrix.diagonal v)] :

⅟ (matrix.diagonal v) = matrix.diagonal (⅟ v)

source

def matrix.invertible_of_diagonal_invertible {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] (v : n → α) [invertible (matrix.diagonal v)] :

invertible v

v is invertible if diagonal v is

Equations

matrix.invertible_of_diagonal_invertible v = {inv_of := (⅟ (matrix.diagonal v)).diag, inv_of_mul_self := _, mul_inv_of_self := _}

source

@[simp]

theorem matrix.diagonal_invertible_equiv_invertible_symm_apply {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] (v : n → α) [invertible v] :

⇑((matrix.diagonal_invertible_equiv_invertible v).symm) _inst_5 = matrix.diagonal_invertible v

source

@[simp]

theorem matrix.diagonal_invertible_equiv_invertible_apply {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] (v : n → α) [invertible (matrix.diagonal v)] :

⇑(matrix.diagonal_invertible_equiv_invertible v) _inst_4 = matrix.invertible_of_diagonal_invertible v

source

def matrix.diagonal_invertible_equiv_invertible {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] (v : n → α) :

invertible (matrix.diagonal v) ≃ invertible v

Together matrix.diagonal_invertible and matrix.invertible_of_diagonal_invertible form an equivalence, although both sides of the equiv are subsingleton anyway.

Equations

matrix.diagonal_invertible_equiv_invertible v = {to_fun := matrix.invertible_of_diagonal_invertible v, inv_fun := matrix.diagonal_invertible v, left_inv := _, right_inv := _}

source

@[simp]

theorem matrix.is_unit_diagonal {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] {v : n → α} :

is_unit (matrix.diagonal v) ↔ is_unit v

When lowered to a prop, matrix.diagonal_invertible_equiv_invertible forms an iff.

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theorem matrix.inv_diagonal {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] (v : n → α) :

(matrix.diagonal v)⁻¹ = matrix.diagonal (ring.inverse v)

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@[simp]

theorem matrix.inv_inv_inv {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] (A : matrix n n α) :

A⁻¹ ⁻¹ ⁻¹ = A⁻¹

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theorem matrix.mul_inv_rev {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] (A B : matrix n n α) :

(A.mul B)⁻¹ = B⁻¹.mul A⁻¹

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theorem matrix.list_prod_inv_reverse {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] (l : list (matrix n n α)) :

(l.prod)⁻¹ = (list.map has_inv.inv l.reverse).prod

A version of list.prod_inv_reverse for matrix.has_inv.

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@[simp]

theorem matrix.det_smul_inv_mul_vec_eq_cramer {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] (A : matrix n n α) (b : n → α) (h : is_unit A.det) :

A.det • A⁻¹.mul_vec b = ⇑(A.cramer) b

One form of Cramer's rule. See matrix.mul_vec_cramer for a stronger form.

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@[simp]

theorem matrix.det_smul_inv_vec_mul_eq_cramer_transpose {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] (A : matrix n n α) (b : n → α) (h : is_unit A.det) :

A.det • matrix.vec_mul b A⁻¹ = ⇑(A.transpose.cramer) b

One form of Cramer's rule. See matrix.mul_vec_cramer for a stronger form.

Inverses of permutated matrices #

Note that the simp-normal form of matrix.reindex is matrix.submatrix, so we prove most of these results about only the latter.

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def matrix.submatrix_equiv_invertible {m : Type u} {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] [fintype m] [decidable_eq m] (A : matrix m m α) (e₁ e₂ : n ≃ m) [invertible A] :

invertible (A.submatrix ⇑e₁ ⇑e₂)

A.submatrix e₁ e₂ is invertible if A is

Equations

A.submatrix_equiv_invertible e₁ e₂ = (A.submatrix ⇑e₁ ⇑e₂).invertible_of_right_inverse ((⅟ A).submatrix ⇑e₂ ⇑e₁) _

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def matrix.invertible_of_submatrix_equiv_invertible {m : Type u} {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] [fintype m] [decidable_eq m] (A : matrix m m α) (e₁ e₂ : n ≃ m) [invertible (A.submatrix ⇑e₁ ⇑e₂)] :

invertible A

A is invertible if A.submatrix e₁ e₂ is

Equations

A.invertible_of_submatrix_equiv_invertible e₁ e₂ = A.invertible_of_right_inverse ((⅟ (A.submatrix ⇑e₁ ⇑e₂)).submatrix ⇑(e₂.symm) ⇑(e₁.symm)) _

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theorem matrix.inv_of_submatrix_equiv_eq {m : Type u} {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] [fintype m] [decidable_eq m] (A : matrix m m α) (e₁ e₂ : n ≃ m) [invertible A] [invertible (A.submatrix ⇑e₁ ⇑e₂)] :

⅟ (A.submatrix ⇑e₁ ⇑e₂) = (⅟ A).submatrix ⇑e₂ ⇑e₁

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def matrix.submatrix_equiv_invertible_equiv_invertible {m : Type u} {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] [fintype m] [decidable_eq m] (A : matrix m m α) (e₁ e₂ : n ≃ m) :

invertible (A.submatrix ⇑e₁ ⇑e₂) ≃ invertible A

Together matrix.submatrix_equiv_invertible and matrix.invertible_of_submatrix_equiv_invertible form an equivalence, although both sides of the equiv are subsingleton anyway.

Equations

A.submatrix_equiv_invertible_equiv_invertible e₁ e₂ = {to_fun := λ (_x : invertible (A.submatrix ⇑e₁ ⇑e₂)), A.invertible_of_submatrix_equiv_invertible e₁ e₂, inv_fun := λ (_x : invertible A), A.submatrix_equiv_invertible e₁ e₂, left_inv := _, right_inv := _}

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@[simp]

theorem matrix.submatrix_equiv_invertible_equiv_invertible_apply {m : Type u} {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] [fintype m] [decidable_eq m] (A : matrix m m α) (e₁ e₂ : n ≃ m) (_x : invertible (A.submatrix ⇑e₁ ⇑e₂)) :

⇑(A.submatrix_equiv_invertible_equiv_invertible e₁ e₂) _x = A.invertible_of_submatrix_equiv_invertible e₁ e₂

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@[simp]

theorem matrix.submatrix_equiv_invertible_equiv_invertible_symm_apply {m : Type u} {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] [fintype m] [decidable_eq m] (A : matrix m m α) (e₁ e₂ : n ≃ m) (_x : invertible A) :

⇑((A.submatrix_equiv_invertible_equiv_invertible e₁ e₂).symm) _x = A.submatrix_equiv_invertible e₁ e₂

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@[simp]

theorem matrix.is_unit_submatrix_equiv {m : Type u} {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] [fintype m] [decidable_eq m] {A : matrix m m α} (e₁ e₂ : n ≃ m) :

is_unit (A.submatrix ⇑e₁ ⇑e₂) ↔ is_unit A

When lowered to a prop, matrix.invertible_of_submatrix_equiv_invertible forms an iff.

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@[simp]

theorem matrix.inv_submatrix_equiv {m : Type u} {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] [fintype m] [decidable_eq m] (A : matrix m m α) (e₁ e₂ : n ≃ m) :

(A.submatrix ⇑e₁ ⇑e₂)⁻¹ = A⁻¹.submatrix ⇑e₂ ⇑e₁

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theorem matrix.inv_reindex {m : Type u} {n : Type u'} {α : Type v} [fintype n] [decidable_eq n] [comm_ring α] [fintype m] [decidable_eq m] (e₁ e₂ : n ≃ m) (A : matrix n n α) :

(⇑(matrix.reindex e₁ e₂) A)⁻¹ = ⇑(matrix.reindex e₂ e₁) A⁻¹