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feat(LinearAlgebra/Matrix/Permutation): permMatrix as a homomorphism #34041

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feat(LinearAlgebra/Matrix/Permutation): permMatrix as a homomorphism #34041

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18 changes: 17 additions & 1 deletion Mathlib/LinearAlgebra/Matrix/Permutation.lean
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Expand Up @@ -31,7 +31,7 @@ This file defines the matrix associated with a permutation

open Equiv

variable {n R : Type*} [DecidableEq n] (σ : Perm n)
variable {n R : Type*} [DecidableEq n] (σ τ : Perm n)

variable (R) in
/-- the permutation matrix associated with an `Equiv.Perm` -/
Expand All @@ -40,6 +40,10 @@ abbrev Equiv.Perm.permMatrix [Zero R] [One R] : Matrix n n R :=

namespace Matrix

@[simp]
lemma permMatrix_one [Zero R] [One R] : (1 : Equiv.Perm n).permMatrix R = 1 := by
simp [← Matrix.ext_iff, Matrix.one_apply]
Comment on lines +43 to +45
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@themathqueen themathqueen Jan 17, 2026 •
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how about also adding the Equiv.refl part?

Suggested change
@[simp]
lemma permMatrix_one [Zero R] [One R] : (1 : Equiv.Perm n).permMatrix R = 1 := by
simp [← Matrix.ext_iff, Matrix.one_apply]
@[simp] lemma permMatrix_refl [Zero R] [One R] : Equiv.Perm.permMatrix R (.refl n) = 1 := by
simp [← Matrix.ext_iff, Matrix.one_apply]
@[simp]
lemma permMatrix_one [Zero R] [One R] : (1 : Equiv.Perm n).permMatrix R = 1 :=
permMatrix_refl


@[simp]
lemma transpose_permMatrix [Zero R] [One R] : (σ.permMatrix R).transpose = (σ⁻¹).permMatrix R := by
rw [← PEquiv.toMatrix_symm, ← Equiv.toPEquiv_symm, ← Equiv.Perm.inv_def]
Expand Down Expand Up @@ -74,6 +78,18 @@ lemma vecMul_permMatrix {v : n → R} [CommRing R] :
ext j
simp [vecMul_eq_sum, Pi.single, Function.update, ← Equiv.symm_apply_eq]

@[simp]
lemma permMatrix_mul [NonAssocSemiring R] :
(σ * τ).permMatrix R = τ.permMatrix R * σ.permMatrix R := by
rw [Perm.permMatrix, Perm.mul_def, toPEquiv_trans, PEquiv.toMatrix_trans]

/-- `permMatrix` as a homomorphism. -/
@[simps]
def permMatrixHom [NonAssocSemiring R] : Perm n →* Matrix n n R where
toFun σ := σ⁻¹.permMatrix R
map_one' := permMatrix_one
map_mul' σ τ := by rw [_root_.mul_inv_rev, permMatrix_mul]

open scoped Matrix.Norms.L2Operator

variable {𝕜 : Type*} [RCLike 𝕜]
Expand Down