| import OAI.Analysis.HyperbolicCones.MatrixLinear |
| import OAI.Analysis.HyperbolicCones.PolynomialEvaluation |
|
|
| namespace OAI |
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| noncomputable section |
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|
| open scoped BigOperators Matrix.Norms.L2Operator MatrixOrder ComplexOrder |
| open Matrix |
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|
| 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 |
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| end Paper256 |
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| end |
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| end OAI |
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