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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 49 50 51 52 53 54 55 56 57 | import Mathlib
import OAI.Probability.BinarySweep.MatrixBounds.Whitening
namespace OAI
noncomputable section
open scoped BigOperators Classical ComplexOrder MatrixOrder
open Matrix
namespace BinaryCoordinateSweeps.Density
variable {I W : Type*} [Fintype I] [DecidableEq I] [Fintype W]
omit [DecidableEq I] in
lemma trace_re_real_smul (a : ℝ) (X : Matrix I I ℂ) :
((a : ℂ) • X).trace.re = a * X.trace.re := by
simp [Matrix.trace_smul,Complex.mul_re]
omit [DecidableEq I] in
lemma trace_re_nat_smul (a : ℕ) (X : Matrix I I ℂ) :
((a : ℂ) • X).trace.re = a * X.trace.re := by
simpa only [Complex.ofReal_natCast] using trace_re_real_smul (a:ℝ) X
lemma trace_density_whitening (R Q P C : Matrix I I ℂ) (d a : ℝ)
(V : W → Matrix I I ℂ) (hR : R.PosSemidef) (hC : C.IsHermitian)
(hcap : (C*Q*C-(d:ℂ)•(Q*P)).PosSemidef)
(hdom : ((a:ℂ)•∑w, V w-Q).PosSemidef) :
d*(R*Q*P).trace.re ≤ a*∑w, (C*R*C*V w).trace.re := by
have h1 := trace_product_re_mono hR hcap
rw [Matrix.mul_smul,trace_re_real_smul,← mul_assoc R Q P] at h1
have h2 := hdom.mul_mul_conjTranspose_same C
rw [hC.eq,Matrix.mul_sub,Matrix.sub_mul,Matrix.mul_smul,Matrix.smul_mul,
Matrix.mul_sum,Matrix.sum_mul] at h2
have h3 := trace_product_re_mono hR h2
rw [Matrix.mul_smul,trace_re_real_smul,Matrix.mul_sum,Matrix.trace_sum,
Complex.re_sum] at h3
have he (w : W) : (R*(C*V w*C)).trace = (C*R*C*V w).trace := by
rw [← mul_assoc R (C*V w) C,Matrix.trace_mul_cycle]
congr 1
noncomm_ring
simp only [he] at h3
exact h1.trans h3
lemma trace_right_density_domination (R S : Matrix I I ℂ) (a : ℝ)
(U : W → Matrix I I ℂ) (hS : S.PosSemidef)
(hdom : ((a:ℂ)•∑w, U w-R).PosSemidef) :
(R*S).trace.re ≤ a * ∑w, (U w*S).trace.re := by
have ht := trace_product_re_mono hS hdom
rw [Matrix.mul_smul,trace_re_real_smul,Matrix.mul_sum,Matrix.trace_sum,Complex.re_sum,
Matrix.trace_mul_comm S R] at ht
simpa only [Matrix.trace_mul_comm S] using ht
end BinaryCoordinateSweeps.Density
end
end OAI
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