| import Mathlib |
|
|
| namespace OAI |
|
|
| section |
| section |
| namespace CAT0Fillings |
| open Matrix |
| open scoped BigOperators |
|
|
| variable {ι : Type*} [Fintype ι] [DecidableEq ι] |
| def totalRankOneForm (H : Matrix ι ι ℝ) (c : ℝ) (u : ι → ℝ) : |
| Matrix (Unit ⊕ ι) (Unit ⊕ ι) ℝ := |
| fromBlocks (0 : Matrix Unit Unit ℝ) 0 0 H+ |
| vecMulVec (Sum.elim (fun _ => c) u) (Sum.elim (fun _ => c) u) |
|
|
| lemma totalRankOneForm_det (H : Matrix ι ι ℝ) (c : ℝ) (u : ι → ℝ) : |
| (totalRankOneForm H c u).det = c^2*H.det := by |
| let E : Matrix (Unit ⊕ ι) (Unit ⊕ ι) ℝ := |
| fromBlocks (Matrix.of fun _ _ => c) 0 (replicateCol Unit u) 1 |
| let D : Matrix (Unit ⊕ ι) (Unit ⊕ ι) ℝ := fromBlocks 1 0 0 H |
| have he : totalRankOneForm H c u = E*D*E.transpose := by |
| dsimp [E,D] |
| rw [Matrix.fromBlocks_transpose,Matrix.fromBlocks_multiply,Matrix.fromBlocks_multiply] |
| simp only [Matrix.mul_one,Matrix.one_mul,Matrix.mul_zero,Matrix.zero_mul, |
| add_zero,zero_add,Matrix.transpose_zero,Matrix.transpose_one] |
| ext i j |
| cases i <;> cases j <;> |
| simp [totalRankOneForm,Matrix.mul_apply,Matrix.vecMulVec_apply, |
| Matrix.replicateCol,Matrix.transpose_apply, mul_comm,add_comm] |
| have hd : D.det = H.det := by simp [D,Matrix.det_fromBlocks_zero₂₁] |
| have hE : E.det = c := by simp [E,Matrix.det_unique] |
| rw [he,Matrix.det_mul,Matrix.det_mul,Matrix.det_transpose,hE,hd] |
| ring |
|
|
| lemma det_sub_vecMulVec (G : Matrix ι ι ℝ) (α : ι → ℝ) (hG : G.det ≠ 0) : |
| (G-vecMulVec α α).det = G.det*(1-α ⬝ᵥ G⁻¹.mulVec α) := by |
| have he : G-vecMulVec α α = |
| G+replicateCol Unit (-α)*replicateRow Unit α := by |
| ext i j |
| simp [Matrix.mul_apply,Matrix.vecMulVec_apply,Matrix.replicateCol,Matrix.replicateRow, |
| sub_eq_add_neg] |
| rw [he,Matrix.det_add_replicateCol_mul_replicateRow (isUnit_iff_ne_zero.mpr hG)] |
| congr 1 |
| rw [Matrix.det_unique] |
| simp only [Matrix.add_apply,Matrix.one_apply_eq,Matrix.mul_apply, |
| Matrix.replicateRow_apply,Matrix.replicateCol_apply,Pi.neg_apply] |
| simp only [Finset.sum_neg_distrib,mul_neg] |
| simp only [dotProduct,Matrix.mulVec,Finset.mul_sum,Finset.sum_mul] |
| rw [Finset.sum_comm] |
| simp only [sub_eq_add_neg,mul_assoc] |
|
|
| omit [DecidableEq ι] in |
|
|
| lemma totalRankOneForm_quadratic (H : Matrix ι ι ℝ) (c : ℝ) (u : ι → ℝ) |
| (v : Unit ⊕ ι → ℝ) : |
| v ⬝ᵥ (totalRankOneForm H c u).mulVec v = |
| (c*v (Sum.inl ()) + u ⬝ᵥ (v ∘ Sum.inr))^2 + |
| (v ∘ Sum.inr) ⬝ᵥ H.mulVec (v ∘ Sum.inr) := by |
| rw [totalRankOneForm,Matrix.add_mulVec,dotProduct_add,Matrix.vecMulVec_mulVec, |
| dotProduct_smul,op_smul_eq_smul,smul_eq_mul,dotProduct_comm v] |
| rw [Matrix.fromBlocks_mulVec] |
| simp only [Matrix.zero_mulVec,zero_add,add_zero] |
| have hsplit : v = Sum.elim (v ∘ Sum.inl) (v ∘ Sum.inr) := by ext x; cases x <;> rfl |
| conv_lhs => rw [hsplit] |
| simp only [sumElim_dotProduct_sumElim] |
| simp [dotProduct,Function.comp_def,pow_two,mul_comm,add_comm] |
| |
| omit [DecidableEq ι] in |
| |
| lemma totalRankOneForm_posSemidef (H : Matrix ι ι ℝ) (c : ℝ) (u : ι → ℝ) |
| (hH : H.PosSemidef) : (totalRankOneForm H c u).PosSemidef := by |
| apply Matrix.PosSemidef.of_dotProduct_mulVec_nonneg |
| · apply Matrix.IsHermitian.add |
| · rw [Matrix.IsHermitian] |
| simpa only [Matrix.fromBlocks_conjTranspose,Matrix.conjTranspose_zero] |
| using congrArg (fun A => fromBlocks (0 : Matrix Unit Unit ℝ) 0 0 A) hH.isHermitian.eq |
| · simpa only [star_trivial] using (Matrix.posSemidef_vecMulVec_self_star |
| (Sum.elim (fun _ => c) u)).isHermitian |
| · intro v |
| simp only [star_trivial,totalRankOneForm_quadratic] |
| exact add_nonneg (sq_nonneg _) (hH.dotProduct_mulVec_nonneg _) |
|
|
| lemma radial_inverse_norm_le_one (G : Matrix ι ι ℝ) (α : ι → ℝ) |
| (hG : G.PosDef) (hα : ∀ v : ι → ℝ, (α ⬝ᵥ v)^2 ≤ v ⬝ᵥ G.mulVec v) : |
| α ⬝ᵥ G⁻¹.mulVec α ≤ 1 := by |
| have hi : G.mulVec (G⁻¹.mulVec α) = α := by |
| rw [Matrix.mulVec_mulVec,Matrix.mul_nonsing_inv G (isUnit_iff_ne_zero.mpr hG.det_pos.ne'), |
| Matrix.one_mulVec] |
| have hh := hα (G⁻¹.mulVec α) |
| rw [hi,dotProduct_comm (G⁻¹.mulVec α)] at hh |
| by_contra! hn |
| have hc := mul_pos (lt_trans zero_lt_one hn) (sub_pos.mpr hn) |
| nlinarith |
|
|
| end CAT0Fillings |
| end |
| end |
|
|
| end OAI |
|
|
