| 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] |
| · rw [hQ i j h] |
|
|
| 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 |
|
|
