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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 #

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] :
A * (A * B) = B

A copy of invOf_mul_self_assoc for rectangular matrices.

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 * (A * B) = B

A copy of mul_invOf_self_assoc for rectangular matrices.

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 * B * B = A

A copy of mul_invOf_mul_self_cancel for rectangular matrices.

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 * B = A

A copy of mul_mul_invOf_self_cancel for rectangular matrices.

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] :

The conjugate transpose of an invertible matrix is invertible.

A matrix is invertible if the conjugate transpose is invertible.

Instances For
    @[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) :
    instance Matrix.invertibleTranspose {n : Type u_2} {α : Type u_3} [Fintype n] [DecidableEq n] [CommSemiring α] (A : Matrix n n α) [Invertible A] :

    The transpose of an invertible matrix is invertible.

    Aᵀ is invertible when A is.

    Instances For

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

      Instances For
        @[simp]
        theorem Matrix.isUnit_transpose {n : Type u_2} {α : Type u_3} [Fintype n] [DecidableEq n] [CommSemiring α] (A : Matrix n n α) :