`BookProof.ChapterConditional.trace_gram_eq_one` (B : Matrix Y X 𝕜) (hB : ∑ x, ∑ y, ‖B y x‖ ^ 2 = 1) : (Bᴴ * B).trace = ((1 : ℝ) : 𝕜)

Proved

BookProof.ChapterConditional.trace_gram_eq_one

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterConditional.

BookProof.ChapterConditional.trace_gram_eq_one (B : Matrix Y X 𝕜) (hB : ∑ x, ∑ y, ‖B y x‖ ^ 2 = 1) : (Bᴴ * B).trace = ((1 : ℝ) : 𝕜)

Formalization note: Lean 4 identifier BookProof.ChapterConditional.trace_gram_eq_one.

Preamble

-- Generated from ChapterConditional.lean — theorem BookProof.ChapterConditional.trace_gram_eq_one
import Mathlib
import Definitions.Def_ChapterConditional
open BookProof.ChapterConditional


open scoped BigOperators Matrix
open Finset


variable {X Y : Type*} [Fintype X] [Fintype Y] [DecidableEq X]
variable {𝕜 : Type*} [RCLike 𝕜]

Formal statement

theorem BookProof.ChapterConditional.trace_gram_eq_one (B : Matrix Y X 𝕜)
    (hB : ∑ x, ∑ y, ‖B y x‖ ^ 2 = 1) :
    (Bᴴ * B).trace = ((1 : ℝ) : 𝕜) := by sorry

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