File size: 2,967 Bytes
9eeb9c7 | 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 70 71 72 73 74 | import OAI.Analysis.HyperbolicCones.MatrixLinear
import OAI.Analysis.HyperbolicCones.PolynomialEvaluation
namespace OAI
noncomputable section
open scoped BigOperators Matrix.Norms.L2Operator MatrixOrder ComplexOrder
open Matrix
namespace Paper256
theorem isHermitian_complexify {n : ℕ} (A : Mat n ℝ) (hA : A.IsHermitian) :
(A.map Complex.ofReal).IsHermitian :=
hA.map _ (by intro x; simp)
theorem complex_quadratic_re {n : ℕ} (A : Mat n ℝ) (w : Fin n → ℂ) :
(dotProduct (star w) (A.map Complex.ofReal *ᵥ w)).re =
dotProduct (fun i => (w i).re) (A *ᵥ fun i => (w i).re) +
dotProduct (fun i => (w i).im) (A *ᵥ fun i => (w i).im) := by
have ht (i j : Fin n) : (star (w i) * ((A i j : ℂ) * w j)).re =
(w i).re * (A i j * (w j).re) + (w i).im * (A i j * (w j).im) := by
simp [Complex.mul_re, Complex.mul_im]
simp only [dotProduct, mulVec, Matrix.map_apply, Pi.star_apply, Finset.mul_sum,
Complex.re_sum, ht, Finset.sum_add_distrib]
theorem posSemidef_complexify {n : ℕ} (A : Mat n ℝ) (hA : A.PosSemidef) :
(A.map Complex.ofReal).PosSemidef := by
have hh := isHermitian_complexify A hA.isHermitian
apply Matrix.PosSemidef.of_dotProduct_mulVec_nonneg hh
intro w
apply RCLike.nonneg_iff.2
refine ⟨?_, hh.im_star_dotProduct_mulVec_self w⟩
rw [show RCLike.re (dotProduct (star w) (A.map Complex.ofReal *ᵥ w)) =
(dotProduct (star w) (A.map Complex.ofReal *ᵥ w)).re from rfl, complex_quadratic_re]
exact add_nonneg (by simpa using hA.dotProduct_mulVec_nonneg (fun i => (w i).re))
(by simpa using hA.dotProduct_mulVec_nonneg (fun i => (w i).im))
theorem posDef_complexify {n : ℕ} (A : Mat n ℝ) (hA : A.PosDef) :
(A.map Complex.ofReal).PosDef := by
have hh := isHermitian_complexify A hA.isHermitian
apply Matrix.PosDef.of_dotProduct_mulVec_pos hh
intro w hw
apply RCLike.pos_iff.2
refine ⟨?_, hh.im_star_dotProduct_mulVec_self w⟩
change 0 < (dotProduct (star w) (A.map Complex.ofReal *ᵥ w)).re
rw [complex_quadratic_re]
by_cases hr : (fun i => (w i).re) = 0
· have hi : (fun i => (w i).im) ≠ 0 := by
intro hi
apply hw
funext i
apply Complex.ext
· exact congrFun hr i
· exact congrFun hi i
exact add_pos_of_nonneg_of_pos
(by simpa using hA.posSemidef.dotProduct_mulVec_nonneg (fun i => (w i).re))
(by simpa using hA.dotProduct_mulVec_pos hi)
· exact add_pos_of_pos_of_nonneg (by simpa using hA.dotProduct_mulVec_pos hr)
(by simpa using hA.posSemidef.dotProduct_mulVec_nonneg (fun i => (w i).im))
theorem complex_resolvent_isUnit {n : ℕ} (A : Mat n ℝ) (hA : A.IsHermitian)
(z : ℂ) (hz : z.im ≠ 0) : IsUnit (z • (1 : Mat n ℂ) - A.map Complex.ofReal) := by
by_contra hn
apply hz
apply (isHermitian_complexify A hA).isSelfAdjoint.im_eq_zero_of_mem_spectrum
simpa only [spectrum.mem_iff, Algebra.algebraMap_eq_smul_one] using hn
end Paper256
end
end OAI
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