| import Mathlib |
| import OAI.Probability.SKGap.Model |
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|
| namespace OAI |
|
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| section |
| noncomputable section |
| open MeasureTheory ProbabilityTheory InformationTheory Real Set |
| open scoped NNReal ENNReal |
| open Filter |
| open scoped Topology |
| noncomputable section |
| open Matrix Real |
| open scoped BigOperators Matrix.Norms.Frobenius ENNReal NNReal |
| noncomputable section |
| open Matrix Real |
| open scoped BigOperators Matrix.Norms.Frobenius NNReal |
| noncomputable section |
| open MeasureTheory ProbabilityTheory Real Set Filter |
| open MeasureTheory.Measure |
| open scoped ENNReal NNReal MeasureTheory Topology |
| open MeasureTheory |
| noncomputable section |
| noncomputable section |
| open MeasureTheory Set NormedSpace |
| open scoped Topology |
| noncomputable section |
| open Matrix Real |
| open scoped BigOperators Matrix.Norms.Frobenius |
| noncomputable section |
| open Set Real |
| open scoped Topology |
| noncomputable section |
| open Matrix Set Filter |
| open scoped Topology Matrix.Norms.Frobenius |
| noncomputable section |
| open Matrix NormedSpace ContinuousLinearMap |
| open scoped Matrix.Norms.Frobenius |
| noncomputable section |
| open Matrix |
| namespace SKGap.ResolventIdentity |
| variable {ι : Type*} [Fintype ι] [DecidableEq ι] |
| variable {𝕜 : Type*} [RCLike 𝕜] |
|
|
| lemma inverse_one_sub_square_mul (D V : Matrix ι ι 𝕜) |
| (hS : IsUnit (1-D*V*D)) : |
| (1-D*D*V)⁻¹ = 1+D*(1-D*V*D)⁻¹*D*V := by |
| apply Matrix.inv_eq_right_inv |
| have hSS : (1-D*V*D)*(1-D*V*D)⁻¹ = 1 := |
| Matrix.mul_nonsing_inv _ (((1-D*V*D).isUnit_iff_isUnit_det).mp hS) |
| calc |
| _ = (1-D*D*V)+D*((1-D*V*D)*(1-D*V*D)⁻¹)*D*V := by noncomm_ring |
| _ = 1 := by rw [hSS]; noncomm_ring |
|
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| lemma inverse_one_sub_square_mul_unit (D V : Matrix ι ι 𝕜) |
| (hS : IsUnit (1-D*V*D)) : IsUnit (1-D*D*V) := by |
| have hSS : (1-D*V*D)*(1-D*V*D)⁻¹ = 1 := |
| Matrix.mul_nonsing_inv _ (((1-D*V*D).isUnit_iff_isUnit_det).mp hS) |
| apply ((1-D*D*V).isUnit_iff_isUnit_det).mpr |
| have hh : (1-D*D*V)*(1+D*(1-D*V*D)⁻¹*D*V) = 1 := by |
| calc |
| _ = (1-D*D*V)+D*((1-D*V*D)*(1-D*V*D)⁻¹)*D*V := by noncomm_ring |
| _ = 1 := by rw [hSS]; noncomm_ring |
| exact Matrix.isUnit_det_of_right_inverse hh |
|
|
| lemma inverse_one_sub_square_mul_square (D V : Matrix ι ι 𝕜) |
| (hS : IsUnit (1-D*V*D)) : |
| (1-D*D*V)⁻¹*(D*D) = D*(1-D*V*D)⁻¹*D := by |
| rw [inverse_one_sub_square_mul D V hS] |
| have hSS : (1-D*V*D)⁻¹*(1-D*V*D) = 1 := |
| Matrix.nonsing_inv_mul _ (((1-D*V*D).isUnit_iff_isUnit_det).mp hS) |
| calc |
| _ = D*(1-D*V*D)⁻¹*D+D*(1-(1-D*V*D)⁻¹*(1-D*V*D))*D := by noncomm_ring |
| _ = _ := by rw [hSS]; simp |
|
|
| end SKGap.ResolventIdentity |
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| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
|
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| end OAI |
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|
