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feat(LinearAlgebra/Matrix): nonsingular inverse commutes with any monoid hom of matrix rings #39733

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feat(LinearAlgebra/Matrix): nonsingular inverse commutes with any monoid hom of matrix rings #39733

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17 changes: 17 additions & 0 deletions Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean
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Expand Up @@ -488,6 +488,23 @@ theorem inv_inj (h : A⁻¹ = B⁻¹) (h' : IsUnit A.det) : A = B := by

end InvEqInv

section Map

variable {R S F : Type*} [CommRing R] [CommRing S]
[FunLike F (Matrix n n R) (Matrix n n S)]
[MonoidHomClass F (Matrix n n R) (Matrix n n S)]

/-- Any monoid homomorphism `f : Matrix n n R →* Matrix n n S` between matrix rings
sends the nonsingular inverse of an invertible matrix to the nonsingular inverse of its image.
In particular this applies to `RingHom`, `RingEquiv`, `AlgHom`, `AlgEquiv`, and `StarAlgEquiv`
between matrix rings, since each of these provides a `MonoidHomClass` instance. -/
theorem map_nonsing_inv (f : F) {A : Matrix n n R} (hA : IsUnit A.det) :
(f A)⁻¹ = f A⁻¹ := by
apply Matrix.inv_eq_right_inv
rw [← map_mul, Matrix.mul_nonsing_inv A hA, map_one]

end Map

variable (A)

set_option backward.isDefEq.respectTransparency false in
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