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9eeb9c7 | 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 | 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
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