`BookProof.DensitySpectral.density_eigenvalues_nonneg` {ρ : Matrix n n ℂ} (h : IsDensityMatrix ρ) (i : n) : 0 ≤ h.1.eigenvalues i

Proved

BookProof.DensitySpectral.density_eigenvalues_nonneg

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterDensitySpectral.

BookProof.DensitySpectral.density_eigenvalues_nonneg {ρ : Matrix n n ℂ} (h : IsDensityMatrix ρ) (i : n) : 0 ≤ h.1.eigenvalues i

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

Preamble

-- Generated from ChapterDensitySpectral.lean — theorem BookProof.DensitySpectral.density_eigenvalues_nonneg
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_eigenvalues_nonneg {ρ : Matrix n n ℂ} (h : IsDensityMatrix ρ) (i : n) :
    0 ≤ h.1.eigenvalues i := by sorry

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