| import Mathlib.Analysis.InnerProductSpace.Projection.FiniteDimensional |
| import Mathlib.Analysis.InnerProductSpace.Adjoint |
| import Mathlib.LinearAlgebra.Trace |
|
|
| namespace OAI |
|
|
| /-! |
| # Traces of actual orthogonal projections |
|
|
| The trace calculation uses the algebraic theorem for a projection onto a |
| submodule. In particular, no spectral decomposition or matrix norm instance |
| is needed. Matrix transport is valid in every basis, and hence in the |
| orthonormal basis used for the weighted grading. |
| -/ |
|
|
| noncomputable section |
|
|
| namespace LeanBlast.GotsmanLinial |
|
|
| section General |
|
|
| variable {𝕜 E ι : Type*} [RCLike 𝕜] [NormedAddCommGroup E] |
| [InnerProductSpace 𝕜 E] [FiniteDimensional 𝕜 E] |
|
|
| /-- The actual orthogonal projection is algebraically a projection onto its |
| specified submodule. -/ |
| theorem starProjection_isProj (K : Submodule 𝕜 E) : |
| LinearMap.IsProj K K.starProjection.toLinearMap where |
| map_mem := K.starProjection_apply_mem |
| map_id := fun _ hx => Submodule.starProjection_eq_self_iff.mpr hx |
|
|
| /-- The trace of the actual orthogonal projection is the dimension of its range. -/ |
| theorem trace_starProjection (K : Submodule 𝕜 E) : |
| LinearMap.trace 𝕜 E K.starProjection.toLinearMap = (Module.finrank 𝕜 K : 𝕜) := |
| (starProjection_isProj K).trace |
|
|
| variable [Fintype ι] [DecidableEq ι] |
|
|
| /-- The matrix trace of an orthogonal projection is independent of the chosen basis. -/ |
| theorem trace_toMatrix_starProjection (b : Module.Basis ι 𝕜 E) (K : Submodule 𝕜 E) : |
| Matrix.trace (LinearMap.toMatrix b b K.starProjection.toLinearMap) = |
| (Module.finrank 𝕜 K : 𝕜) := by |
| rw [← LinearMap.trace_eq_matrix_trace 𝕜 b, trace_starProjection] |
|
|
| /-- The orthonormal matrix representation used by the grading has the correct trace. -/ |
| theorem trace_toMatrixOrthonormal_starProjection (b : OrthonormalBasis ι 𝕜 E) |
| (K : Submodule 𝕜 E) : |
| Matrix.trace (LinearMap.toMatrixOrthonormal b K.starProjection.toLinearMap) = |
| (Module.finrank 𝕜 K : 𝕜) := |
| trace_toMatrix_starProjection b.toBasis K |
|
|
| end General |
|
|
| section Complex |
|
|
| variable {E ι : Type*} [NormedAddCommGroup E] [InnerProductSpace ℂ E] |
| [FiniteDimensional ℂ E] [Fintype ι] [DecidableEq ι] |
|
|
| /-- The real trace required by Hilbert--Schmidt block-mass computations. -/ |
| theorem re_trace_toMatrixOrthonormal_starProjection (b : OrthonormalBasis ι ℂ E) |
| (K : Submodule ℂ E) : |
| (Matrix.trace (LinearMap.toMatrixOrthonormal b K.starProjection.toLinearMap)).re = |
| (Module.finrank ℂ K : ℝ) := by |
| rw [trace_toMatrixOrthonormal_starProjection] |
| simp |
|
|
| end Complex |
|
|
| end LeanBlast.GotsmanLinial |
|
|
| end |
|
|
| end OAI |
|
|
