`BookProof.DensitySpectral.density_diag_eq_kernel_apply` (U : Matrix n n ℂ) (d : n → ℝ) (i : n) : (U * Matrix.diagonal (RCLike.ofReal ∘ d) * Uᴴ : Matrix n n ℂ) i i = ((∑ k, bornKer

Proved

BookProof.DensitySpectral.density_diag_eq_kernel_apply

by leonardopedro · Oct 7, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterDensityMarginalConditional.

BookProof.DensitySpectral.density_diag_eq_kernel_apply (U : Matrix n n ℂ) (d : n → ℝ) (i : n) : (U * Matrix.diagonal (RCLike.ofReal ∘ d) * Uᴴ : Matrix n n ℂ) i i = ((∑ k, bornKernel U i k * d k : ℝ) : ℂ)

Formalization note: Lean 4 identifier BookProof.DensitySpectral.density_diag_eq_kernel_apply.

Preamble

-- Generated from ChapterDensityMarginalConditional.lean — theorem BookProof.DensitySpectral.density_diag_eq_kernel_apply
import Mathlib
import Definitions.Def_ChapterDensityMarginalConditional
open BookProof.DensitySpectral



open Matrix
open scoped BigOperators ComplexOrder

variable {n : Type*} [Fintype n] [DecidableEq n]

Formal statement

theorem BookProof.DensitySpectral.density_diag_eq_kernel_apply (U : Matrix n n ℂ) (d : n → ℝ) (i : n) :
    (U * Matrix.diagonal (RCLike.ofReal ∘ d) * Uᴴ : Matrix n n ℂ) i i
      = ((∑ k, bornKernel U i k * d k : ℝ) : ℂ) := by sorry

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