| import OAI.Analysis.HyperbolicCones.KernelSpectrum |
|
|
| namespace OAI |
|
|
| /-! Simultaneous spectral comparison of a positive matrix and its scalar shifts. -/ |
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| noncomputable section |
| open scoped Matrix.Norms.L2Operator MatrixOrder |
| open Matrix |
| namespace Paper256 |
|
|
| theorem matrix_cfc_affine {n : ℕ} (H : Sym n) (a b : ℝ) : |
| cfc (fun x : ℝ => a + b * x) (H : Mat n ℝ) = |
| a • (1 : Mat n ℝ) + b • (H : Mat n ℝ) := by |
| calc |
| _ = algebraMap ℝ (Mat n ℝ) a + cfc (fun x : ℝ => b * x) (H : Mat n ℝ) := |
| cfc_const_add a (fun x : ℝ => b * x) (H : Mat n ℝ) |
| (H.val.finite_real_spectrum.continuousOn _) H.property |
| _ = _ := by |
| have hmul : cfc (fun x : ℝ => b * x) (H : Mat n ℝ) = b • (H : Mat n ℝ) := |
| cfc_const_mul_id b (H : Mat n ℝ) H.property |
| rw [hmul, Algebra.algebraMap_eq_smul_one] |
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|
| theorem posDef_inverse_eq_cfc {n : ℕ} (H : Sym n) (hH : (H : Mat n ℝ).PosDef) : |
| (H : Mat n ℝ)⁻¹ = cfc (fun x : ℝ => x⁻¹) (H : Mat n ℝ) := by |
| have hi := cfc_inv (fun x : ℝ => x) (H : Mat n ℝ) |
| (fun x hx => (posDef_real_spectrum_positive hH hx).ne') |
| (H.val.finite_real_spectrum.continuousOn _) H.property |
| have hid : cfc (fun x : ℝ => x) (H : Mat n ℝ) = (H : Mat n ℝ) := |
| cfc_id' ℝ (H : Mat n ℝ) H.property |
| rw [hid, ← Matrix.nonsing_inv_eq_ringInverse] at hi |
| exact hi.symm |
|
|
| theorem posDef_shift_inverse_eq_cfc {n : ℕ} (H : Sym n) |
| (hH : (H : Mat n ℝ).PosDef) (t : ℝ) (ht : 0 ≤ t) : |
| ((H : Mat n ℝ) + t • 1)⁻¹ = |
| cfc (fun x : ℝ => (x + t)⁻¹) (H : Mat n ℝ) := by |
| have hi := cfc_inv (fun x : ℝ => x + t) (H : Mat n ℝ) |
| (fun x hx => (add_pos_of_pos_of_nonneg (posDef_real_spectrum_positive hH hx) ht).ne') |
| (H.val.finite_real_spectrum.continuousOn _) H.property |
| have hshift : cfc (fun x : ℝ => x + t) (H : Mat n ℝ) = |
| (H : Mat n ℝ) + t • 1 := by |
| simpa only [add_comm, one_mul, one_smul] using matrix_cfc_affine H t 1 |
| rw [hshift, ← Matrix.nonsing_inv_eq_ringInverse] at hi |
| exact hi.symm |
|
|
| theorem posDef_inverse_sub_shift_inverse_posSemidef {n : ℕ} (H : Sym n) |
| (hH : (H : Mat n ℝ).PosDef) (t : ℝ) (ht : 0 ≤ t) : |
| ((H : Mat n ℝ)⁻¹ - ((H : Mat n ℝ) + t • 1)⁻¹).PosSemidef := by |
| apply Matrix.nonneg_iff_posSemidef.mp |
| rw [sub_nonneg, posDef_inverse_eq_cfc H hH, posDef_shift_inverse_eq_cfc H hH t ht] |
| apply cfc_mono _ (H.val.finite_real_spectrum.continuousOn _) |
| (H.val.finite_real_spectrum.continuousOn _) |
| intro x hx |
| exact inv_anti₀ (posDef_real_spectrum_positive hH hx) (le_add_of_nonneg_right ht) |
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| end Paper256 |
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| end |
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| end OAI |
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