File size: 2,644 Bytes
e320e82 | 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 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
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