File size: 2,404 Bytes
ea76aac | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 | import Mathlib
import OAI.Computability.MaxCut.Model
namespace OAI
/-!
Equal kernels give the same orbit under invertible postcomposition.
The canonical range equivalence comes from the first isomorphism theorem;
Mathlib's finite-dimensional subspace extension theorem then extends it to an
automorphism of the ambient codomain. No orbit or counting premise is assumed.
-/
namespace MaxCutGames.Inverse.KMSKernelOrbits
noncomputable section
variable {K E F : Type*} [DivisionRing K]
[AddCommGroup E] [Module K E] [AddCommGroup F] [Module K F]
/-- The range identification induced by a common kernel. -/
def rangeEquivOfKerEq (S T : F →ₗ[K] E) (hker : S.ker = T.ker) :
S.range ≃ₗ[K] T.range :=
S.quotKerEquivRange.symm ≪≫ₗ
Submodule.quotEquivOfEq S.ker T.ker hker ≪≫ₗ T.quotKerEquivRange
/-- This identification respects every original vector, including vectors in
the kernel and maps of rank zero. -/
theorem rangeEquivOfKerEq_apply (S T : F →ₗ[K] E) (hker : S.ker = T.ker) (x : F) :
(rangeEquivOfKerEq S T hker ⟨S x, ⟨x, rfl⟩⟩ : E) = T x := by
change (T.quotKerEquivRange ((Submodule.quotEquivOfEq S.ker T.ker hker)
(S.quotKerEquivRange.symm ⟨S x, ⟨x, rfl⟩⟩)) : E) = T x
have hs : (⟨S x, ⟨x, rfl⟩⟩ : S.range) =
S.quotKerEquivRange (Submodule.Quotient.mk x) := rfl
rw [hs, LinearEquiv.symm_apply_apply]
rfl
/-- Finite-dimensional image suffices: the ambient codomain and domain may
otherwise be infinite-dimensional. -/
theorem exists_postcomp_equiv_of_ker_eq (S T : F →ₗ[K] E)
[FiniteDimensional K S.range] (hker : S.ker = T.ker) :
∃ g : E ≃ₗ[K] E, g.toLinearMap.comp S = T := by
obtain ⟨g, hg⟩ := Submodule.exists_linearEquiv_restrict_eq
(rangeEquivOfKerEq S T hker)
refine ⟨g, ?_⟩
ext x
change g (S x) = T x
exact (hg ⟨S x, ⟨x, rfl⟩⟩).symm.trans (rangeEquivOfKerEq_apply S T hker x)
/-- Invertible postcomposition never changes the kernel. Thus the kernel is
an exact orbit invariant, rather than only a necessary condition. -/
theorem ker_eq_iff_exists_postcomp_equiv (S T : F →ₗ[K] E)
[FiniteDimensional K S.range] :
S.ker = T.ker ↔ ∃ g : E ≃ₗ[K] E, g.toLinearMap.comp S = T := by
constructor
· exact exists_postcomp_equiv_of_ker_eq S T
· rintro ⟨g, rfl⟩
ext x
simp
end
end MaxCutGames.Inverse.KMSKernelOrbits
end OAI
|