`BookProof.ChapterSchurIrreducible.isIrreducible_iff_schur` (M : System ℂ V) (hM : M.IsNormal) : M.IsIrreducible ↔ ∀ S : V →L[ℂ] V, M.Commutes S → IsSelfAdjoint S → ∃ c : ℂ, S = c

Proved

BookProof.ChapterSchurIrreducible.isIrreducible_iff_schur

by leonardopedro · Oct 17, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterSchurIrreducible.

BookProof.ChapterSchurIrreducible.isIrreducible_iff_schur (M : System ℂ V) (hM : M.IsNormal) : M.IsIrreducible ↔ ∀ S : V →L[ℂ] V, M.Commutes S → IsSelfAdjoint S → ∃ c : ℂ, S = c • (1 : V →L[ℂ] V)

Formalization note: Lean 4 identifier BookProof.ChapterSchurIrreducible.isIrreducible_iff_schur.

Preamble

-- Generated from ChapterSchurIrreducible.lean — theorem BookProof.ChapterSchurIrreducible.isIrreducible_iff_schur
import Mathlib
import Definitions.Def_ChapterSchurIrreducible
import Definitions.Def_ChapterA
open BookProof.ChapterA
open BookProof.ChapterA
open BookProof.ChapterSchurIrreducible


open scoped ComplexConjugate InnerProductSpace


open BookProof.ChapterA BookProof.ChapterA

variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [CompleteSpace V]

Formal statement

theorem BookProof.ChapterSchurIrreducible.isIrreducible_iff_schur (M : System ℂ V) (hM : M.IsNormal) :
    M.IsIrreducible ↔
      ∀ S : V →L[ℂ] V, M.Commutes S → IsSelfAdjoint S → ∃ c : ℂ, S = c • (1 : V →L[ℂ] V) := by sorry

View graph