`BookProof.DensitySpectral.density_spectral` {ρ : Matrix n n ℂ} (h : IsDensityMatrix ρ) : ρ = (h.1.eigenvectorUnitary : Matrix n n ℂ) * diagonal (RCLike.ofReal ∘ h.1.eigenvalues) *

Proved

BookProof.DensitySpectral.density_spectral

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterDensitySpectral.

BookProof.DensitySpectral.density_spectral {ρ : Matrix n n ℂ} (h : IsDensityMatrix ρ) : ρ = (h.1.eigenvectorUnitary : Matrix n n ℂ) * diagonal (RCLike.ofReal ∘ h.1.eigenvalues) * (h.1.eigenvectorUnitary : Matrix n n ℂ)ᴴ

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

Preamble

-- Generated from ChapterDensitySpectral.lean — theorem BookProof.DensitySpectral.density_spectral
import Mathlib
import Definitions.Def_ChapterDensitySpectral
import Definitions.Def_ChapterB4
import Definitions.Def_ChapterDensityMarginalConditional
open BookProof.ChapterB4
open BookProof.DensitySpectral



open Matrix
open scoped BigOperators ComplexOrder

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

Formal statement

theorem BookProof.DensitySpectral.density_spectral {ρ : Matrix n n ℂ} (h : IsDensityMatrix ρ) :
    ρ = (h.1.eigenvectorUnitary : Matrix n n ℂ)
        * diagonal (RCLike.ofReal ∘ h.1.eigenvalues)
        * (h.1.eigenvectorUnitary : Matrix n n ℂ)ᴴ := by sorry

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