Documentation

Mathlib.Data.Matrix.Invertible

Extra lemmas about invertible matrices #

A few of the Invertible lemmas generalize to multiplication of rectangular matrices.

For lemmas about the matrix inverse in terms of the determinant and adjugate, see Matrix.inv in LinearAlgebra/Matrix/NonsingularInverse.lean.

Main results #

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theorem Matrix.invOf_mul_self_assoc {m : Type u_1} {n : Type u_2} {α : Type u_3} [Fintype n] [DecidableEq n] [Semiring α] (A : Matrix n n α) (B : Matrix n m α) [Invertible A] :
⅟A * (A * B) = B

A copy of invOf_mul_self_assoc for rectangular matrices.

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theorem Matrix.mul_invOf_self_assoc {m : Type u_1} {n : Type u_2} {α : Type u_3} [Fintype n] [DecidableEq n] [Semiring α] (A : Matrix n n α) (B : Matrix n m α) [Invertible A] :
A * (⅟A * B) = B

A copy of mul_invOf_self_assoc for rectangular matrices.

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theorem Matrix.mul_invOf_mul_self_cancel {m : Type u_1} {n : Type u_2} {α : Type u_3} [Fintype n] [DecidableEq n] [Semiring α] (A : Matrix m n α) (B : Matrix n n α) [Invertible B] :
A * ⅟B * B = A

A copy of mul_invOf_mul_self_cancel for rectangular matrices.

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theorem Matrix.mul_mul_invOf_self_cancel {m : Type u_1} {n : Type u_2} {α : Type u_3} [Fintype n] [DecidableEq n] [Semiring α] (A : Matrix m n α) (B : Matrix n n α) [Invertible B] :
A * B * ⅟B = A

A copy of mul_mul_invOf_self_cancel for rectangular matrices.

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instance Matrix.invertibleConjTranspose {n : Type u_2} {α : Type u_3} [Fintype n] [DecidableEq n] [Semiring α] [StarRing α] (A : Matrix n n α) [Invertible A] :
Invertible A.conjTranspose

The conjugate transpose of an invertible matrix is invertible.

Equations
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theorem Matrix.conjTranspose_invOf {n : Type u_2} {α : Type u_3} [Fintype n] [DecidableEq n] [Semiring α] [StarRing α] (A : Matrix n n α) [Invertible A] [Invertible A.conjTranspose] :
(⅟A).conjTranspose = ⅟A.conjTranspose
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def Matrix.invertibleOfInvertibleConjTranspose {n : Type u_2} {α : Type u_3} [Fintype n] [DecidableEq n] [Semiring α] [StarRing α] (A : Matrix n n α) [Invertible A.conjTranspose] :
Invertible A

A matrix is invertible if the conjugate transpose is invertible.

Equations
  • A.invertibleOfInvertibleConjTranspose = ⋯.mpr (⋯.mpr inferInstance)
Instances For
    source
    @[simp]
    theorem Matrix.isUnit_conjTranspose {n : Type u_2} {α : Type u_3} [Fintype n] [DecidableEq n] [Semiring α] [StarRing α] (A : Matrix n n α) :
    IsUnit A.conjTranspose ↔ IsUnit A
    source
    instance Matrix.invertibleTranspose {n : Type u_2} {α : Type u_3} [Fintype n] [DecidableEq n] [CommSemiring α] (A : Matrix n n α) [Invertible A] :
    Invertible A.transpose

    The transpose of an invertible matrix is invertible.

    Equations
    • A.invertibleTranspose = { invOf := (⅟A).transpose, invOf_mul_self := ⋯, mul_invOf_self := ⋯ }
    source
    theorem Matrix.transpose_invOf {n : Type u_2} {α : Type u_3} [Fintype n] [DecidableEq n] [CommSemiring α] (A : Matrix n n α) [Invertible A] [Invertible A.transpose] :
    (⅟A).transpose = ⅟A.transpose
    source
    def Matrix.invertibleOfInvertibleTranspose {n : Type u_2} {α : Type u_3} [Fintype n] [DecidableEq n] [CommSemiring α] (A : Matrix n n α) [Invertible A.transpose] :
    Invertible A

    Aᵀ is invertible when A is.

    Equations
    • A.invertibleOfInvertibleTranspose = { invOf := (⅟A.transpose).transpose, invOf_mul_self := ⋯, mul_invOf_self := ⋯ }
    Instances For
      source
      @[simp]
      theorem Matrix.transposeInvertibleEquivInvertible_apply {n : Type u_2} {α : Type u_3} [Fintype n] [DecidableEq n] [CommSemiring α] (A : Matrix n n α) [Invertible A.transpose] :
      A.transposeInvertibleEquivInvertible inst✝ = A.invertibleOfInvertibleTranspose
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      @[simp]
      theorem Matrix.transposeInvertibleEquivInvertible_symm_apply {n : Type u_2} {α : Type u_3} [Fintype n] [DecidableEq n] [CommSemiring α] (A : Matrix n n α) [Invertible A] :
      A.transposeInvertibleEquivInvertible.symm inst✝ = A.invertibleTranspose
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      def Matrix.transposeInvertibleEquivInvertible {n : Type u_2} {α : Type u_3} [Fintype n] [DecidableEq n] [CommSemiring α] (A : Matrix n n α) :
      Invertible A.transpose ≃ Invertible A

      Together Matrix.invertibleTranspose and Matrix.invertibleOfInvertibleTranspose form an equivalence, although both sides of the equiv are subsingleton anyway.

      Equations
      • One or more equations did not get rendered due to their size.
      Instances For
        source
        @[simp]
        theorem Matrix.isUnit_transpose {n : Type u_2} {α : Type u_3} [Fintype n] [DecidableEq n] [CommSemiring α] (A : Matrix n n α) :
        IsUnit A.transpose ↔ IsUnit A