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9b6e014 | 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 | import Mathlib
namespace OAI
noncomputable section
open scoped BigOperators Classical ComplexOrder MatrixOrder
open Matrix
namespace BinaryCoordinateSweeps.Density
variable {I J : Type*} [Fintype I] [DecidableEq I] [DecidableEq J]
lemma block_diagonal_commute (f : I → J) (Q : Matrix I I ℂ)
(hQ : ∀ i j, f i ≠ f j → Q i j = 0) (a : J) :
Commute Q (Matrix.diagonal (fun i => if f i = a then (1 : ℂ) else 0)) := by
ext i j
simp only [Matrix.mul_diagonal, Matrix.diagonal_mul]
by_cases h : f i = f j
· rw [h]; ring
· rw [hQ i j h]; ring
lemma sqrt_block_diagonal (f : I → J) (Q : Matrix I I ℂ)
(hQ : ∀ i j, f i ≠ f j → Q i j = 0) :
∀ i j, f i ≠ f j → CFC.sqrt Q i j = 0 := by
intro i j hij
have h := (block_diagonal_commute f Q hQ (f i)).cfc_nnreal NNReal.sqrt
rw [← CFC.sqrt_eq_cfc] at h
have hh := congrFun (congrFun h.eq i) j
simp only [Matrix.mul_diagonal, Matrix.diagonal_mul, ite_true,
ite_eq_right (Ne.symm hij), mul_zero, one_mul] at hh
exact hh.symm
lemma sqrt_submatrix_invariant (Q : Matrix I I ℂ) (hQ : Q.PosSemidef)
(e : Equiv.Perm I) (he : Q.submatrix e e = Q) :
(CFC.sqrt Q).submatrix e e = CFC.sqrt Q := by
apply Eq.symm
apply CFC.sqrt_unique _ ((CFC.sqrt_nonneg Q).posSemidef.submatrix e).nonneg
rw [Matrix.submatrix_mul_equiv, CFC.sqrt_mul_sqrt_self Q hQ.nonneg, he]
lemma sqrt_gram (Q : Matrix I I ℂ) (hQ : Q.PosSemidef) :
CFC.sqrt Q * (CFC.sqrt Q).conjTranspose = Q := by
rw [(CFC.sqrt_nonneg Q).posSemidef.1.eq, CFC.sqrt_mul_sqrt_self Q hQ.nonneg]
end BinaryCoordinateSweeps.Density
end
end OAI
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