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 (Matrix.conjTranspose A)

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 (Matrix.conjTranspose A)] :
Matrix.conjTranspose ⅟A = ⅟(Matrix.conjTranspose A)
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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 (Matrix.conjTranspose A)] :
Invertible A

A matrix is invertible if the conjugate transpose is invertible.

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    @[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 (Matrix.conjTranspose A) ↔ IsUnit A
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    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 (Matrix.transpose A)

    The transpose of an invertible matrix is invertible.

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    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 (Matrix.transpose A)] :
    Matrix.transpose ⅟A = ⅟(Matrix.transpose A)
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    def Matrix.invertibleOfInvertibleTranspose {n : Type u_2} {α : Type u_3} [Fintype n] [DecidableEq n] [CommSemiring α] (A : Matrix n n α) [Invertible (Matrix.transpose A)] :
    Invertible A

    Aᵀ is invertible when A is.

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      @[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 (Matrix.transpose A)] :
      ↑(Matrix.transposeInvertibleEquivInvertible A) inst✝ = Matrix.invertibleOfInvertibleTranspose A
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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] :
      ↑(Matrix.transposeInvertibleEquivInvertible A).symm inst✝ = Matrix.invertibleTranspose A
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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 (Matrix.transpose A) ≃ Invertible A

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

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        @[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 (Matrix.transpose A) ↔ IsUnit A