| import OAI.Analysis.HyperbolicCones.MatrixLinear | |
| namespace OAI | |
| noncomputable section | |
| open Matrix | |
| universe u v | |
| namespace Paper256 | |
| theorem posSemidef_fromBlocks_diagonal_iff {ι : Type u} {κ : Type v} [Fintype ι] [Fintype κ] | |
| (A : Matrix ι ι ℝ) (B : Matrix κ κ ℝ) : | |
| (Matrix.fromBlocks A 0 0 B).PosSemidef ↔ A.PosSemidef ∧ B.PosSemidef := by | |
| constructor | |
| · intro h | |
| constructor | |
| · convert h.submatrix Sum.inl using 1 | |
| ext i j | |
| rfl | |
| · convert h.submatrix Sum.inr using 1 | |
| ext i j | |
| rfl | |
| · rintro ⟨hA, hB⟩ | |
| apply Matrix.PosSemidef.of_dotProduct_mulVec_nonneg | |
| (Matrix.IsHermitian.fromBlocks hA.isHermitian (by simp) hB.isHermitian) | |
| intro v | |
| have ha := hA.dotProduct_mulVec_nonneg (v ∘ Sum.inl) | |
| have hb := hB.dotProduct_mulVec_nonneg (v ∘ Sum.inr) | |
| simpa [Matrix.fromBlocks_mulVec, dotProduct, Fintype.sum_sum_type] using add_nonneg ha hb | |
| theorem matrix_compression_posSemidef_iff {ι : Type u} [Fintype ι] | |
| (p : ι → Prop) [DecidablePred p] (M : Matrix ι ι ℝ) (hM : M.IsHermitian) | |
| (hzero : ∀ i, ¬p i → ∀ j, M i j = 0) : | |
| M.PosSemidef ↔ (M.submatrix (Subtype.val : {i // p i} → ι) Subtype.val).PosSemidef := by | |
| let e := Equiv.sumCompl p | |
| have heq : M.submatrix e e = Matrix.fromBlocks | |
| (M.submatrix (Subtype.val : {i // p i} → ι) Subtype.val) 0 0 0 := by | |
| ext (i | i) (j | j) | |
| · rfl | |
| · change M i.val j.val = 0 | |
| have hs : M i.val j.val = M j.val i.val := by simpa using hM.apply j.val i.val | |
| rw [hs, hzero j.val j.property] | |
| · exact hzero i.val i.property j.val | |
| · exact hzero i.val i.property j.val | |
| calc | |
| M.PosSemidef ↔ (M.submatrix e e).PosSemidef := (Matrix.posSemidef_submatrix_equiv e).symm | |
| _ ↔ (M.submatrix (Subtype.val : {i // p i} → ι) Subtype.val).PosSemidef := by | |
| rw [heq, posSemidef_fromBlocks_diagonal_iff] | |
| simp only [Matrix.PosSemidef.zero, and_true] | |
| end Paper256 | |
| end | |
| end OAI | |