| import OAI.Analysis.DirectCrouzeix.NumericalRange | |
| namespace OAI | |
| universe u_1 u_2 u_3 u_4 u_5 u_6 u_7 | |
| noncomputable section | |
| open scoped Matrix Matrix.Norms.L2Operator Kronecker | |
| namespace DirectCrouzeix | |
| open scoped MatrixOrder ComplexOrder | |
| def hsSq {ι : Type u_1} {κ : Type u_2} [Fintype ι] [Fintype κ] (M : Matrix ι κ ℂ) : ℝ := | |
| (Matrix.trace (Mᴴ * M)).re | |
| theorem hsSq_nonneg {ι : Type u_3} {κ : Type u_4} [Fintype ι] [Fintype κ] | |
| (M : Matrix ι κ ℂ) : 0 ≤ hsSq M := by | |
| exact (Complex.nonneg_iff.mp (Matrix.posSemidef_conjTranspose_mul_self M).trace_nonneg).1 | |
| theorem re_trace_mul_nonneg {ι : Type u_5} [Fintype ι] [DecidableEq ι] | |
| {A B : Matrix ι ι ℂ} (hA : A.PosSemidef) (hB : B.PosSemidef) : | |
| 0 ≤ (Matrix.trace (A * B)).re := by | |
| obtain ⟨C, hC⟩ := CStarAlgebra.nonneg_iff_eq_star_mul_self.mp hA.nonneg | |
| rw [hC, Matrix.star_eq_conjTranspose, Matrix.trace_mul_cycle, Matrix.trace_mul_cycle] | |
| exact (Complex.nonneg_iff.mp (hB.mul_mul_conjTranspose_same C).trace_nonneg).1 | |
| theorem re_trace_block_comparison {ι : Type u_6} [Fintype ι] [DecidableEq ι] | |
| (D J : Matrix ι ι ℂ) (hD : D.PosSemidef) (hJ : J.IsHermitian) : | |
| 0 ≤ (Matrix.trace (D * J * D * J)).re := by | |
| have hp := re_trace_mul_nonneg hD (hD.mul_mul_conjTranspose_same J) | |
| simpa only [hJ.eq, Matrix.mul_assoc] using hp | |
| theorem positive_block_hsSq {ι : Type u_7} [Fintype ι] [DecidableEq ι] | |
| {U V Z : Matrix ι ι ℂ} | |
| (h : (Matrix.fromBlocks U Zᴴ Z V).PosSemidef) : | |
| 2 * hsSq Z ≤ hsSq U + hsSq V := by | |
| let D := Matrix.fromBlocks U Zᴴ Z V | |
| let J : Matrix (Sum ι ι) (Sum ι ι) ℂ := Matrix.fromBlocks 1 0 0 (-1) | |
| have hJ : J.IsHermitian := by | |
| exact Matrix.IsHermitian.fromBlocks Matrix.isHermitian_one (by simp) | |
| Matrix.isHermitian_one.neg | |
| have hU : U.IsHermitian := by | |
| have hh := h.isHermitian | |
| exact hh.submatrix Sum.inl | |
| have hV : V.IsHermitian := by | |
| have hh := h.isHermitian | |
| exact hh.submatrix Sum.inr | |
| have hp := re_trace_block_comparison D J h hJ | |
| have trace_blocks (A B C E : Matrix ι ι ℂ) : | |
| Matrix.trace (Matrix.fromBlocks A B C E) = Matrix.trace A + Matrix.trace E := by | |
| simp [Matrix.trace] | |
| have he : Matrix.trace (D * J * D * J) = | |
| Matrix.trace (U * U) + Matrix.trace (V * V) - | |
| 2 * Matrix.trace (Zᴴ * Z) := by | |
| simp [D, J, Matrix.fromBlocks_multiply, trace_blocks, | |
| Matrix.trace_mul_comm Z Zᴴ] | |
| ring | |
| rw [he] at hp | |
| norm_num at hp | |
| simpa only [hsSq, hU.eq, hV.eq] using hp | |
| end DirectCrouzeix | |
| end | |
| end OAI | |