feat(Analysis/InnerProductSpace): generalized determinant of a rectangle matrix / linear map

Author: wwylele

Mathlib.lean

@@ -1946,6 +1946,7 @@ public import Mathlib.Analysis.InnerProductSpace.LinearMap
 public import Mathlib.Analysis.InnerProductSpace.LinearPMap
 public import Mathlib.Analysis.InnerProductSpace.MeanErgodic
 public import Mathlib.Analysis.InnerProductSpace.MulOpposite
+public import Mathlib.Analysis.InnerProductSpace.NormDet
 public import Mathlib.Analysis.InnerProductSpace.NormPow
 public import Mathlib.Analysis.InnerProductSpace.OfNorm
 public import Mathlib.Analysis.InnerProductSpace.Orientation

Mathlib/Analysis/InnerProductSpace/GramMatrix.lean

@@ -8,6 +8,7 @@ module
 public import Mathlib.Analysis.InnerProductSpace.Basic
 public import Mathlib.Analysis.InnerProductSpace.PiL2
 public import Mathlib.LinearAlgebra.Matrix.PosDef
+import Mathlib.Analysis.Matrix.Order
 
 /-! # Gram Matrices
 
@@ -95,6 +96,12 @@ theorem linearIndependent_of_posDef_gram {v : n → E} (h_gram : PosDef (gram 
   have := h_gram.dotProduct_mulVec_pos (x := y)
   simp_all [star_dotProduct_gram_mulVec]
 
+omit [Finite n] in
+theorem linearIndependent_of_det_gram_ne_zero [Fintype n] [DecidableEq n] {v : n → E}
+    (h : (gram 𝕜 v).det ≠ 0) : LinearIndependent 𝕜 v := by
+  apply linearIndependent_of_posDef_gram
+  simpa [(Matrix.posSemidef_gram _ _).posDef_iff_isUnit, Matrix.isUnit_iff_isUnit_det] using h
+
 end SemiInnerProductSpace
 
 section NormedInnerProductSpace

Mathlib/Analysis/InnerProductSpace/NormDet.lean

@@ -0,0 +1,356 @@
+/-
+Copyright (c) 2026 Weiyi Wang. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Weiyi Wang
+-/
+module
+
+public import Mathlib.Analysis.InnerProductSpace.Adjoint
+public import Mathlib.Analysis.InnerProductSpace.GramMatrix
+
+/-!
+# Normed determinant of a linear map
+
+Given a rectangular matrix $T$, it is common to talk about $\sqrt{det(T^{H}T)}$, where $T^{H}$ is
+the conjugate transpose of $T$, as a generalization to the determinant of a square matrix. It is the
+$m$-dimensional volume factor for $\mathbb{R}^m \to \mathbb{R}^n$. It is given various names in
+literature:
+* "Jacobian" (definition 3.4 of [lawrenceronald2025]), in the context of volume factor
+  for non-linear map. However, we choose to reserve this name for the matrix consists of
+  derivatives.
+* "Gram determinant", which is already used by `Matrix.gram`, and it is often referring to the
+  $det(T^{H}T)$ without the square root.
+* "Nonnegative determinant" (definition 1 of [haruoyoshiohidetoki2006]).
+
+Without a standardized name, we give a descriptive name `LinearMap.normDet` to reflect its
+definition and show that it is a generalization of `‖(f : LinearMap 𝕜 U U).det‖`
+(See `LinearMap.normDet_eq_norm_det`). We also construct this on linear maps between inner product
+spaces instead of matrices, and allow the codomain to have infinite dimension.
+
+## Main definition
+* `LinearMap.normDet` : the norm determinant of a linear map.
+
+## Main result
+* `ContinuousLinearMap.normDet_sq` and `LinearMap.normDet_sq`: The square of `f.normDet`
+  equals to the determinant of `f.adjoint ∘ₗ f`.
+* `LinearMap.normDet_sq_eq_det_gram`: The square of `f.normDet` equals to the determinant of the
+  gram matrix formed by vectors mapped from an orthonormal basis.
+
+## TODO
+
+Show that `LinearMap.normDet` is the product of all singular values of the map.
+-/
+
+public section
+
+namespace LinearMap
+
+variable {𝕜 U V W : Type*} [RCLike 𝕜] [NormedAddCommGroup U] [InnerProductSpace 𝕜 U]
+  [FiniteDimensional 𝕜 U] [NormedAddCommGroup V] [InnerProductSpace 𝕜 V] [NormedAddCommGroup W]
+  [InnerProductSpace 𝕜 W]
+
+open Classical in
+/--
+The norm determinant of a linear map `f : U →ₗ[𝕜] V` is defined as the norm of the determinant of
+the square matrix from `U →ₗ[𝕜] f.range` using a pair of orthonormal basis of equal dimensions.
+(See `LinearMap.normDet_eq` for using arbitrary orthonormal basis)
+
+If such basis doesn't exist (i.e. the map is not injective), the norm determinant is zero.
+(See `LinearMap.normDet_eq_zero_iff_ker_ne_bot`)
+-/
+noncomputable def normDet (f : U →ₗ[𝕜] V) : ℝ :=
+  if h : Nonempty (OrthonormalBasis (Fin (Module.finrank 𝕜 U)) 𝕜 f.range) then
+    ‖(f.rangeRestrict.toMatrix (stdOrthonormalBasis 𝕜 U).toBasis h.some.toBasis).det‖
+  else
+    0
+
+theorem normDet_nonneg (f : U →ₗ[𝕜] V) : 0 ≤ f.normDet := by
+  unfold normDet
+  split <;> simp
+
+/--
+`LinearMap.normDet` is well-defined under any pair of orthonormal basis.
+-/
+theorem normDet_eq {ι : Type*} [Fintype ι] [DecidableEq ι] (f : U →ₗ[𝕜] V)
+    (bu : OrthonormalBasis ι 𝕜 U) (bv : OrthonormalBasis ι 𝕜 f.range) :
+    f.normDet = ‖(f.rangeRestrict.toMatrix bu.toBasis bv.toBasis).det‖ := by
+  have hrank : Module.finrank 𝕜 U = Module.finrank 𝕜 f.range := by
+    rw [Module.finrank_eq_nat_card_basis bu.toBasis, Module.finrank_eq_nat_card_basis bv.toBasis]
+  have h : Nonempty (OrthonormalBasis (Fin (Module.finrank 𝕜 U)) 𝕜 f.range) := by
+    rw [hrank]
+    exact ⟨stdOrthonormalBasis 𝕜 f.range⟩
+  simp only [normDet, h, ↓reduceDIte]
+  rw [← basis_toMatrix_mul_linearMap_toMatrix_mul_basis_toMatrix (stdOrthonormalBasis 𝕜 U).toBasis
+    bu.toBasis h.some.toBasis bv.toBasis]
+  have h1 : bu.toBasis.toMatrix (stdOrthonormalBasis 𝕜 U).toBasis *
+      (stdOrthonormalBasis 𝕜 U).toBasis.toMatrix bu.toBasis = 1 :=
+    Module.Basis.toMatrix_mul_toMatrix_flip _ _
+  have h2 : (stdOrthonormalBasis 𝕜 U).toBasis.toMatrix bu.toBasis *
+      bu.toBasis.toMatrix (stdOrthonormalBasis 𝕜 U).toBasis = 1 :=
+    Module.Basis.toMatrix_mul_toMatrix_flip _ _
+  rw [← Matrix.det_comm' h1 h2, ← Matrix.mul_assoc, Matrix.det_mul, norm_mul]
+  suffices ‖(bu.toBasis.toMatrix (stdOrthonormalBasis 𝕜 U).toBasis *
+      h.some.toBasis.toMatrix ⇑bv.toBasis).det‖ = 1 by
+    rw [this]
+    simp
+  apply CStarRing.norm_of_mem_unitary
+  apply Matrix.det_of_mem_unitary
+  rw [Matrix.mem_unitaryGroup_iff, Matrix.star_eq_conjTranspose, Matrix.conjTranspose_mul,
+    ← Matrix.mul_assoc, Matrix.mul_assoc (bu.toBasis.toMatrix (stdOrthonormalBasis 𝕜 U).toBasis)]
+  simp
+
+/--
+`LinearMap.normDet` vanishes iff the map is not injective.
+-/
+theorem normDet_eq_zero_iff_ker_ne_bot {f : U →ₗ[𝕜] V} :
+    f.normDet = 0 ↔ f.ker ≠ ⊥ where
+  mp h := by
+    contrapose! h
+    let g : U ≃ₗ[𝕜] f.range := LinearEquiv.ofBijective f.rangeRestrict
+      ⟨by simpa using ker_eq_bot.mp h, f.surjective_rangeRestrict⟩
+    let bu := stdOrthonormalBasis 𝕜 U
+    let bv := g.finrank_eq.symm ▸ stdOrthonormalBasis 𝕜 f.range
+    rw [f.normDet_eq bu bv, norm_eq_zero.ne]
+    suffices (f.rangeRestrict.adjoint.toMatrix bv.toBasis bu.toBasis).det *
+        (f.rangeRestrict.toMatrix bu.toBasis bv.toBasis).det ≠ 0  by
+      simpa [toMatrix_adjoint, Matrix.det_conjTranspose] using this
+    simpa [← Matrix.det_mul, ← LinearMap.toMatrix_comp, det_eq_zero_iff_ker_ne_bot,
+      LinearMap.ker_adjoint_comp_self] using h
+  mpr h := by
+    suffices ¬ Nonempty (OrthonormalBasis (Fin (Module.finrank 𝕜 U)) 𝕜 f.range) by
+      simp [normDet, this]
+    contrapose! h
+    obtain ⟨b⟩ := h
+    have hrank : Module.finrank 𝕜 f.range = Module.finrank 𝕜 U := by
+      simpa using Module.finrank_eq_card_basis b.toBasis
+    simpa [hrank] using f.finrank_range_add_finrank_ker
+
+/--
+`LinearMap.normDet` equals to the norm of `LinearMap.det` for an endomorphism.
+-/
+theorem normDet_eq_norm_det (f : U →ₗ[𝕜] U) : f.normDet = ‖f.det‖ := by
+  by_cases h : f.range = ⊤
+  · let bv : OrthonormalBasis (Fin (Module.finrank 𝕜 U)) 𝕜 f.range :=
+      (stdOrthonormalBasis 𝕜 U).map (LinearIsometryEquiv.ofTop U _ h).symm
+    rw [normDet_eq f (stdOrthonormalBasis 𝕜 U) bv]
+    simp [bv, toMatrix_map_right]
+  · have hb : ¬ Nonempty (OrthonormalBasis (Fin (Module.finrank 𝕜 U)) 𝕜 f.range) := by
+      contrapose! h
+      obtain ⟨b⟩ := h
+      apply Submodule.eq_top_of_finrank_eq
+      conv_lhs => rw [Module.finrank_eq_nat_card_basis b.toBasis]
+      conv_rhs => rw [Module.finrank_eq_nat_card_basis (stdOrthonormalBasis 𝕜 U).toBasis]
+    suffices f.det = 0 by
+      simp [this, normDet, hb]
+    simpa [det_eq_zero_iff_ker_ne_bot] using ker_eq_bot_iff_range_eq_top.not.mpr h
+
+/--
+`LinearMap.normDet` of a linear isometry is 1.
+-/
+theorem _root_.LinearIsometry.normDet_eq_one (f : U →ₗᵢ[𝕜] V) : f.toLinearMap.normDet = 1 := by
+  have hrank : Module.finrank 𝕜 U = Module.finrank 𝕜 f.range :=
+    f.equivRange.toLinearEquiv.finrank_eq
+  let b := (stdOrthonormalBasis 𝕜 f.toLinearMap.range)
+  rw [← hrank] at b
+  rw [normDet_eq _ (stdOrthonormalBasis 𝕜 U) b]
+  apply CStarRing.norm_of_mem_unitary
+  exact Matrix.det_of_mem_unitary <| (f.equivRange).toMatrix_mem_unitaryGroup _ _
+
+@[simp]
+theorem normDet_id : (id : U →ₗ[𝕜] U).normDet = 1 :=
+  LinearIsometry.id.normDet_eq_one
+
+@[simp]
+theorem normDet_subtype (p : Submodule 𝕜 U) : p.subtype.normDet = 1 :=
+  p.subtypeₗᵢ.normDet_eq_one
+
+@[simp]
+theorem normDet_of_subsingleton [Subsingleton U] (f : U →ₗ[𝕜] V) : f.normDet = 1 := by
+  have h : Module.finrank 𝕜 U = 0 := Module.finrank_zero_iff.mpr ‹_›
+  have hf : Module.finrank 𝕜 f.range = 0 := by
+    apply Nat.eq_zero_of_le_zero
+    rw [← h]
+    apply LinearMap.finrank_range_le
+  let bu : OrthonormalBasis (Fin 0) 𝕜 U := (stdOrthonormalBasis 𝕜 U).reindex (by rw [h])
+  let bv : OrthonormalBasis (Fin 0) 𝕜 f.range :=
+    (stdOrthonormalBasis 𝕜 f.range).reindex (by rw [hf])
+  rw [normDet_eq _ bu bv]
+  simp
+
+@[simp]
+theorem normDet_zero : (0 : U →ₗ[𝕜] V).normDet = 0 ^ Module.finrank 𝕜 U := by
+  nontriviality U
+  simp [zero_pow Module.finrank_pos.ne.symm, normDet_eq_zero_iff_ker_ne_bot]
+
+@[simp]
+theorem normDet_smul (f : U →ₗ[𝕜] V) (c : 𝕜) :
+    (c • f).normDet = ‖c‖ ^ Module.finrank 𝕜 U * f.normDet := by
+  by_cases hc : c = 0
+  · nontriviality U
+    simp [hc, zero_pow Module.finrank_pos.ne.symm]
+  by_cases h : f.ker = ⊥
+  · have hrank : Module.finrank 𝕜 f.range = Module.finrank 𝕜 U := by
+      obtain hrank := f.finrank_range_add_finrank_ker
+      rw [h] at hrank
+      simpa [h] using hrank
+    let bu : OrthonormalBasis (Fin (Module.finrank 𝕜 U)) 𝕜 U := stdOrthonormalBasis 𝕜 U
+    let bv : OrthonormalBasis (Fin (Module.finrank 𝕜 U)) 𝕜 f.range :=
+      (stdOrthonormalBasis 𝕜 f.range).reindex (by rw [hrank])
+    let bv' : OrthonormalBasis (Fin (Module.finrank 𝕜 U)) 𝕜 (c • f).range := bv.map
+      (LinearIsometryEquiv.ofEq _ _ (LinearMap.range_smul _ _ hc).symm)
+    rw [f.normDet_eq bu bv, (c • f).normDet_eq bu bv', ← norm_pow, ← norm_mul]
+    have : Module.finrank 𝕜 U = Fintype.card (Fin (Module.finrank 𝕜 U)) := by simp
+    conv in c ^ Module.finrank 𝕜 U => rw [this]
+    rw [← Matrix.det_smul, ← map_smul]
+    rfl
+  · have h' : (c • f).ker ≠ ⊥ := by simpa [f.ker_smul _ hc] using h
+    simp [normDet_eq_zero_iff_ker_ne_bot.mpr h, normDet_eq_zero_iff_ker_ne_bot.mpr h']
+
+@[simp]
+theorem normDet_neg (f : U →ₗ[𝕜] V) : (-f).normDet = f.normDet := by
+  simpa using f.normDet_smul (-1)
+
+set_option backward.isDefEq.respectTransparency false in
+/--
+The square of `f.normDet` equals to the determinant of `f.adjoint ∘L f`.
+-/
+theorem _root_.ContinuousLinearMap.normDet_sq [CompleteSpace V] (f : U →L[𝕜] V) :
+    haveI : CompleteSpace U := FiniteDimensional.complete 𝕜 U
+    ↑(f.normDet ^ 2) = (f.adjoint ∘L f).det := by
+  have : CompleteSpace U := FiniteDimensional.complete 𝕜 U
+  have : CompleteSpace f.range := FiniteDimensional.complete 𝕜 f.range
+  let bu := stdOrthonormalBasis 𝕜 U
+  classical
+  by_cases hrank : Module.finrank 𝕜 U = Module.finrank 𝕜 f.range
+  · let b : OrthonormalBasis (Fin (Module.finrank 𝕜 U)) 𝕜 f.range :=
+      (stdOrthonormalBasis 𝕜 f.range).reindex (by
+      rw [← hrank, Module.finrank_eq_card_basis bu.toBasis])
+    have hf : f = f.range.subtypeₗᵢ.toContinuousLinearMap ∘L f.rangeRestrict := rfl
+    conv_rhs => rw [hf]
+    rw [ContinuousLinearMap.adjoint_comp, ← ContinuousLinearMap.comp_assoc,
+      ContinuousLinearMap.comp_assoc (ContinuousLinearMap.adjoint _)]
+    suffices f.range.subtypeₗᵢ.toContinuousLinearMap.adjoint ∘L
+        f.range.subtypeₗᵢ.toContinuousLinearMap = ContinuousLinearMap.id 𝕜 _ by
+      rw [this, ContinuousLinearMap.comp_id, ContinuousLinearMap.det, ContinuousLinearMap.coe_comp,
+        ← det_toMatrix bu.toBasis, toMatrix_comp bu.toBasis b.toBasis bu.toBasis]
+      rw [show (ContinuousLinearMap.adjoint f.rangeRestrict).toLinearMap =
+        f.rangeRestrict.toLinearMap.adjoint by rfl]
+      rw [toMatrix_adjoint]
+      simp [f.toLinearMap.normDet_eq bu b, RCLike.conj_mul]
+    exact f.range.subtypeₗᵢ.adjoint_comp_self
+  · have hb : ¬ Nonempty (OrthonormalBasis (Fin (Module.finrank 𝕜 U)) 𝕜 f.range) := by
+      contrapose! hrank with h
+      obtain ⟨bv⟩ := h
+      rw [Module.finrank_eq_card_basis bu.toBasis, Module.finrank_eq_card_basis bv.toBasis]
+    trans 0
+    · simp [normDet, hb]
+    symm
+    rw [det_eq_zero_iff_ker_ne_bot, ContinuousLinearMap.ker_adjoint_comp_self]
+    contrapose! hrank
+    exact (finrank_range_of_inj <| ker_eq_bot.mp hrank).symm
+
+/--
+The square of `f.normDet` equals to the determinant of `f.adjoint ∘ₗ f` when the codomain is finite
+dimensional.
+-/
+theorem normDet_sq [FiniteDimensional 𝕜 V] (f : U →ₗ[𝕜] V) :
+    ↑(f.normDet ^ 2) = (f.adjoint ∘ₗ f).det := by
+  have : CompleteSpace V := FiniteDimensional.complete 𝕜 V
+  exact f.toContinuousLinearMap.normDet_sq
+
+/--
+The square of `f.normDet` equals to the determinant of the gram matrix formed by vectors mapped from
+an orthonormal basis.
+-/
+theorem normDet_sq_eq_det_gram {ι : Type*} [Fintype ι] [DecidableEq ι] (f : U →ₗ[𝕜] V)
+    (b : OrthonormalBasis ι 𝕜 U) :
+    ↑(f.normDet ^ 2) = (Matrix.gram 𝕜 (f <| b ·)).det := by
+  suffices ↑(f.normDet ^ 2) = (Matrix.gram 𝕜 (f.rangeRestrict <| b ·)).det by
+    simpa
+  by_cases hrank : Module.finrank 𝕜 U = Module.finrank 𝕜 f.range
+  · let bv : OrthonormalBasis ι 𝕜 f.range := (stdOrthonormalBasis 𝕜 f.range).reindex
+      (Fintype.equivFinOfCardEq (by rw [← Module.finrank_eq_card_basis b.toBasis, hrank])).symm
+    rw [Matrix.gram_eq_conjTranspose_mul bv, Matrix.det_mul, Matrix.det_conjTranspose]
+    rw [RCLike.star_def, RCLike.conj_mul, f.normDet_eq b bv]
+    simp only [map_pow]
+    congr
+    ext i j
+    simp [LinearMap.toMatrix_apply]
+  · have hb : ¬ Nonempty (OrthonormalBasis (Fin (Module.finrank 𝕜 U)) 𝕜 f.range) := by
+      contrapose! hrank with h
+      obtain ⟨bv⟩ := h
+      simp [Module.finrank_eq_card_basis bv.toBasis]
+    symm
+    suffices (Matrix.gram 𝕜 (f.rangeRestrict <| b ·)).det = 0 by
+      simpa [normDet, hb]
+    contrapose! hrank with h0
+    obtain hindep := Matrix.linearIndependent_of_det_gram_ne_zero h0
+    apply le_antisymm ?_ (finrank_range_le f)
+    rw [Module.finrank_eq_card_basis b.toBasis]
+    exact hindep.fintype_card_le_finrank
+
+theorem normDet_comp (f : U →ₗ[𝕜] V) (g : V →ₗ[𝕜] W) :
+    (g ∘ₗ f).normDet = (g.domRestrict f.range).normDet * f.normDet := by
+  by_cases hrank : Module.finrank 𝕜 U = Module.finrank 𝕜 f.range
+  · let bu : OrthonormalBasis (Fin (Module.finrank 𝕜 U)) 𝕜 U := stdOrthonormalBasis 𝕜 U
+    let bv : OrthonormalBasis (Fin (Module.finrank 𝕜 U)) 𝕜 f.range :=
+      (stdOrthonormalBasis 𝕜 f.range).reindex (by
+      rw [← hrank, Module.finrank_eq_card_basis bu.toBasis])
+    by_cases hrank' : Module.finrank 𝕜 U = Module.finrank 𝕜 (g ∘ₗ f).range
+    · let bw : OrthonormalBasis (Fin (Module.finrank 𝕜 U)) 𝕜 (g ∘ₗ f).range :=
+        (stdOrthonormalBasis 𝕜 (g ∘ₗ f).range).reindex (by
+        rw [← hrank', Module.finrank_eq_card_basis bu.toBasis])
+      let bw' : OrthonormalBasis (Fin (Module.finrank 𝕜 U)) 𝕜 (g.domRestrict f.range).range :=
+        bw.map (LinearIsometryEquiv.ofEq _ _ (by simp [LinearMap.range_comp]))
+      rw [(g ∘ₗ f).normDet_eq bu bw, f.normDet_eq bu bv, (g.domRestrict f.range).normDet_eq bv bw']
+      rw [← norm_mul, ← Matrix.det_mul, ← LinearMap.toMatrix_comp]
+      rfl
+    · have hker : (g ∘ₗ f).ker ≠ ⊥ := by
+        contrapose! hrank' with hbot
+        obtain h := (g ∘ₗ f).finrank_range_add_finrank_ker.symm
+        rw [hbot] at h
+        simpa using h
+      have hker' : (g.domRestrict f.range).ker ≠ ⊥ := by
+        contrapose! hrank' with hbot
+        obtain h := (g.domRestrict f.range).finrank_range_add_finrank_ker.symm
+        rw [hbot, range_domRestrict] at h
+        rw [hrank, h, range_comp]
+        simp
+      simp [normDet_eq_zero_iff_ker_ne_bot.mpr hker, normDet_eq_zero_iff_ker_ne_bot.mpr hker']
+  · have hker : f.ker ≠ ⊥ := by
+      contrapose! hrank with hbot
+      obtain h := f.finrank_range_add_finrank_ker.symm
+      rw [hbot] at h
+      simpa using h
+    have hker' : (g ∘ₗ f).ker ≠ ⊥ := by
+      contrapose! hker with hbot
+      simpa [hbot] using ker_le_ker_comp f g
+    simp [normDet_eq_zero_iff_ker_ne_bot.mpr hker, normDet_eq_zero_iff_ker_ne_bot.mpr hker']
+
+theorem normDet_comp_of_finrank_eq [FiniteDimensional 𝕜 V] (f : U →ₗ[𝕜] V) (g : V →ₗ[𝕜] W)
+    (h : Module.finrank 𝕜 U = Module.finrank 𝕜 V) :
+    (g ∘ₗ f).normDet = g.normDet * f.normDet := by
+  by_cases htop : f.range = ⊤
+  · rw [normDet_comp]
+    congrm ?_ * _
+    suffices (g.domRestrict f.range).normDet * (id : V →ₗ[𝕜] V).normDet = g.normDet by simpa
+    have : f.range = id.range := by simp [htop]
+    convert (normDet_comp LinearMap.id g).symm
+  · have hker : f.ker ≠ ⊥ := by
+      simpa [ker_eq_bot_iff_range_eq_top_of_finrank_eq_finrank h] using htop
+    have hker' : (g ∘ₗ f).ker ≠ ⊥ := by
+      contrapose! hker with hbot
+      simpa [hbot] using ker_le_ker_comp f g
+    simp [normDet_eq_zero_iff_ker_ne_bot.mpr hker, normDet_eq_zero_iff_ker_ne_bot.mpr hker']
+
+@[simp]
+theorem normDet_codRestrict (p : Submodule 𝕜 V) (f : U →ₗ[𝕜] V) (h : ∀ c, f c ∈ p) :
+    (f.codRestrict p h).normDet = f.normDet := by
+  have : f = p.subtype ∘ₗ f.codRestrict p h := rfl
+  conv_rhs => rw [this]
+  rw [normDet_comp]
+  have : (p.subtype.domRestrict (codRestrict p f h).range).normDet = 1 :=
+    (p.subtypeₗᵢ.comp (codRestrict p f h).range.subtypeₗᵢ).normDet_eq_one
+  simp [this]
+
+end LinearMap