| import Mathlib |
| import OAI.Geometry.PrescribedRicci.TraceCauchySchwarz |
|
|
| namespace OAI |
|
|
| /-! Chern Lu Gradient. -/ |
|
|
| noncomputable section |
| open Matrix |
| open scoped ComplexOrder MatrixOrder |
| namespace MongeAmpere |
| variable {n m : Type*} [Fintype n] [DecidableEq n] [Fintype m] [DecidableEq m] |
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|
| omit [DecidableEq n] [DecidableEq m] in |
| lemma tracePair_sum (B K : Matrix n n ℂ) (W : m → Matrix n n ℂ) (v : m → ℂ) : |
| (B*(∑ i, v i • W i)ᴴ*K*(∑ j, v j • W j)).trace = |
| ∑ i, ∑ j, star (v i)*v j*(B*(W i)ᴴ*K*W j).trace := by |
| simp only [conjTranspose_sum,conjTranspose_smul,Matrix.sum_mul, |
| Matrix.mul_smul,Matrix.smul_mul,trace_sum,trace_smul,smul_eq_mul,Finset.mul_sum] |
| rw [Finset.sum_comm] |
| apply Finset.sum_congr rfl |
| intro i _ |
| apply Finset.sum_congr rfl |
| intro j _ |
| ring |
|
|
| omit [DecidableEq n] [DecidableEq m] in |
| lemma trace_sum_smul (B : Matrix n n ℂ) (W : m → Matrix n n ℂ) (v : m → ℂ) : |
| (B*(∑ i, v i • W i)).trace = ∑ i, v i*(B*W i).trace := by |
| simp only [Matrix.mul_sum,Matrix.mul_smul,trace_sum,trace_smul,smul_eq_mul] |
|
|
| omit [DecidableEq m] in |
| lemma trace_schur_posSemidef (B G : Matrix n n ℂ) (hB : B.PosSemidef) |
| (hG : G.PosDef) (W : m → Matrix n n ℂ) : |
| Matrix.PosSemidef (fun i j => ((B*G).trace.re : ℂ)*(B*(W i)ᴴ*G⁻¹*W j).trace - |
| star (B*W i).trace*(B*W j).trace : Matrix m m ℂ) := by |
| let S := (B*G).trace.re |
| let N : Matrix m m ℂ := fun i j => (S:ℂ)*(B*(W i)ᴴ*G⁻¹*W j).trace - |
| star (B*W i).trace*(B*W j).trace |
| have hs (i j : m) : star (B*(W j)ᴴ*G⁻¹*W i).trace = (B*(W i)ᴴ*G⁻¹*W j).trace := |
| (tracePairCore B G⁻¹ hB hG.inv.posSemidef).conj_inner_symm _ _ |
| apply Matrix.posSemidef_iff_dotProduct_mulVec.mpr |
| constructor |
| · apply Matrix.IsHermitian.ext |
| intro i j |
| change star (N j i) = N i j |
| dsimp only [N] |
| rw [star_sub,star_mul,star_mul,star_star,hs] |
| simp only [Complex.star_def,Complex.conj_ofReal] |
| ring |
| · intro v |
| let C := ∑ i, v i • W i |
| have he : star v ⬝ᵥ (N *ᵥ v) = (S:ℂ)*(B*Cᴴ*G⁻¹*C).trace - |
| star (B*C).trace*(B*C).trace := by |
| dsimp only [C] |
| rw [tracePair_sum,trace_sum_smul] |
| simp only [dotProduct,mulVec,Pi.star_apply,N,Finset.mul_sum,Finset.sum_mul, |
| Finset.sum_sub_distrib,mul_sub,sub_mul,star_sum,star_mul] |
| congr 1 |
| · apply Finset.sum_congr rfl |
| intro i _ |
| apply Finset.sum_congr rfl |
| intro j _ |
| ring |
| · rw [Finset.sum_comm] |
| apply Finset.sum_congr rfl |
| intro i _ |
| apply Finset.sum_congr rfl |
| intro j _ |
| ring |
| change 0 ≤ star v ⬝ᵥ (N *ᵥ v) |
| rw [he] |
| have hq := (tracePairCore B G⁻¹ hB hG.inv.posSemidef).conj_inner_symm C C |
| have hqi : (B*Cᴴ*G⁻¹*C).trace.im = 0 := by |
| have hi := congrArg Complex.im hq |
| simp only [Complex.conj_im] at hi |
| change -(B*Cᴴ*G⁻¹*C).trace.im = (B*Cᴴ*G⁻¹*C).trace.im at hi |
| linarith |
| have hCS := trace_cauchySchwarz B G C hB hG |
| apply Complex.nonneg_iff.mpr |
| constructor |
| · simp only [Complex.sub_re,Complex.mul_re,Complex.ofReal_re,Complex.ofReal_im,zero_mul,sub_zero, |
| Complex.star_def,Complex.conj_re,Complex.conj_im] |
| rw [Complex.sq_norm,Complex.normSq_apply] at hCS |
| dsimp only [S] |
| nlinarith |
| · simp only [Complex.sub_im,Complex.mul_im,Complex.ofReal_re,Complex.ofReal_im,zero_mul,add_zero, |
| hqi,mul_zero,Complex.star_def,Complex.conj_re,Complex.conj_im] |
| ring |
|
|
| lemma chernLu_gradient_bound (B G : Matrix n n ℂ) (hB : B.PosSemidef) |
| (hG : G.PosDef) (W : n → Matrix n n ℂ) : |
| (∑ a, ∑ b, B a b*star (B*W b).trace*(B*W a).trace).re ≤ |
| (B*G).trace.re * (∑ a, ∑ b, B a b*(B*(W b)ᴴ*G⁻¹*W a).trace).re := by |
| have hn := EllipticKernel.trace_mul_nonneg hB (trace_schur_posSemidef B G hB hG W) |
| simp only [Matrix.trace,Matrix.diag_apply,Matrix.mul_apply] at hn |
| have he : (∑ a, ∑ b, B a b * |
| (((B*G).trace.re:ℂ)*(B*(W b)ᴴ*G⁻¹*W a).trace-star (B*W b).trace*(B*W a).trace)) = |
| ((B*G).trace.re:ℂ)*(∑ a, ∑ b, B a b*(B*(W b)ᴴ*G⁻¹*W a).trace) - |
| (∑ a, ∑ b, B a b*star (B*W b).trace*(B*W a).trace) := by |
| simp only [Finset.mul_sum,← Finset.sum_sub_distrib] |
| apply Finset.sum_congr rfl |
| intro a _ |
| apply Finset.sum_congr rfl |
| intro b _ |
| ring |
| change 0 ≤ (∑ a, ∑ b, B a b * |
| (((B*G).trace.re:ℂ)*(B*(W b)ᴴ*G⁻¹*W a).trace-star (B*W b).trace*(B*W a).trace)).re at hn |
| rw [he,Complex.sub_re,Complex.re_ofReal_mul] at hn |
| linarith |
|
|
| end MongeAmpere |
|
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| end |
|
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| end OAI |
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