`BookProof.ChapterA3.bilC_conj` (A M : Matrix (Fin 4) (Fin 4) ℂ) (x : Fin 4 → ℂ) : bilC (Aᵀ * M * A) x = bilC M (fun i => ∑ j, A i j * x j)

Proved

BookProof.ChapterA3.bilC_conj

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterA3h.

BookProof.ChapterA3.bilC_conj (A M : Matrix (Fin 4) (Fin 4) ℂ) (x : Fin 4 → ℂ) : bilC (Aᵀ * M * A) x = bilC M (fun i => ∑ j, A i j * x j)

Formalization note: Lean 4 identifier BookProof.ChapterA3.bilC_conj.

Preamble

-- Generated from ChapterA3h.lean — theorem BookProof.ChapterA3.bilC_conj
import Mathlib
import Definitions.Def_ChapterA3h
import Definitions.Def_ChapterA3
open BookProof.ChapterA3


open Matrix
open scoped ComplexConjugate

Formal statement

theorem BookProof.ChapterA3.bilC_conj (A M : Matrix (Fin 4) (Fin 4) ℂ) (x : Fin 4 → ℂ) :
    bilC (Aᵀ * M * A) x = bilC M (fun i => ∑ j, A i j * x j) := by sorry

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