openAI-math-dataset / lean /OAI /Probability /BinarySweep /MatrixBounds /PositiveTraceRestriction.lean
| 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 | |