File size: 2,550 Bytes
ddc84c5 | 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 59 60 61 62 63 64 65 66 67 68 69 | 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
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