File size: 2,539 Bytes
9b6e014 | 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 65 66 67 | import Mathlib
namespace OAI
noncomputable section
open scoped BigOperators Classical ComplexOrder
namespace BinaryCoordinateSweeps
variable {W : Type*} [NormedAddCommGroup W] [InnerProductSpace ℂ W]
[FiniteDimensional ℂ W]
lemma positive_pow {B : W →ₗ[ℂ] W} (hB : B.IsPositive) (q : ℕ) : (B^q).IsPositive :=
match q with
| 0 => LinearMap.isPositive_one
| 1 => by simpa using hB
| n+2 => by
rw [pow_succ,pow_succ']
simpa only [Module.End.mul_eq_comp,LinearMap.comp_assoc,hB.adjoint_eq] using (positive_pow hB n).conj_adjoint B
theorem positive_trace_restrict_le (S : Submodule ℂ W) (B : W →ₗ[ℂ] W)
(hB : B.IsPositive) (hS : ∀ x ∈ S, B x ∈ S) :
(LinearMap.trace ℂ S (B.restrict hS)).re ≤ (LinearMap.trace ℂ W B).re := by
let P := S.starProjection.toLinearMap
let Q : W →ₗ[ℂ] W := 1-P
have hPsy : P.IsSymmetric := (Submodule.isSymmetricProjection_starProjection S).isSymmetric
have hQsy : Q.IsSymmetric := LinearMap.IsSymmetric.id.sub hPsy
have hPP : P*P = P := by
ext x
exact congrArg (fun t : W →L[ℂ] W => t x) S.isIdempotentElem_starProjection
have hQQ : Q*Q = Q := by
dsimp [Q]
noncomm_ring [hPP]
have hQBQ : (Q*B*Q).IsPositive := by
simpa only [Module.End.mul_eq_comp,LinearMap.comp_assoc,hQsy.adjoint_eq] using hB.conj_adjoint Q
have hpRange : ∀ x, ((P*B)*P) x ∈ S := by
intro x
exact S.starProjection_apply_mem _
have hpRestr : ((P*B)*P).restrict (fun x _ => hpRange x) = B.restrict hS := by
ext x
change P (B (P x.val)) = B x.val
rw [show P x.val = x.val from Submodule.starProjection_eq_self_iff.mpr x.property]
exact Submodule.starProjection_eq_self_iff.mpr (hS x.val x.property)
have hTrP : LinearMap.trace ℂ W (P*B*P) = LinearMap.trace ℂ W (B*P) := by
change LinearMap.trace ℂ W (P.comp (B*P)) = _
rw [LinearMap.trace_comp_comm']
change LinearMap.trace ℂ W (B*P*P) = _
rw [mul_assoc,hPP]
have hTrQ : LinearMap.trace ℂ W (Q*B*Q) = LinearMap.trace ℂ W B -
LinearMap.trace ℂ W (B*P) := by
change LinearMap.trace ℂ W (Q.comp (B*Q)) = _
rw [LinearMap.trace_comp_comm']
change LinearMap.trace ℂ W (B*Q*Q) = _
rw [mul_assoc,hQQ]
dsimp [Q]
rw [mul_sub,mul_one,map_sub]
have hTrace := LinearMap.trace_restrict_eq_of_forall_mem S (P*B*P) hpRange
rw [hpRestr,hTrP] at hTrace
have hn := (Complex.nonneg_iff.mp hQBQ.trace_nonneg).1
rw [hTrQ,Complex.sub_re,← hTrace] at hn
linarith
end BinaryCoordinateSweeps
end
end OAI
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