`BookProof.ChapterNumericalRangeCrouzeix.pearcyVec_apply` (A : E →L[ℂ] E) (c : ℂ) (n : ℕ) (y : E) : pearcyVec A c n y - c • A (pearcyVec A c n y) = y - c ^ n • (A ^ n) y

Proved

BookProof.ChapterNumericalRangeCrouzeix.pearcyVec_apply

by leonardopedro · Oct 10, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterNumericalRangeCrouzeix.

BookProof.ChapterNumericalRangeCrouzeix.pearcyVec_apply (A : E →L[ℂ] E) (c : ℂ) (n : ℕ) (y : E) : pearcyVec A c n y - c • A (pearcyVec A c n y) = y - c ^ n • (A ^ n) y

Formalization note: Lean 4 identifier BookProof.ChapterNumericalRangeCrouzeix.pearcyVec_apply.

Preamble

-- Generated from ChapterNumericalRangeCrouzeix.lean — theorem BookProof.ChapterNumericalRangeCrouzeix.pearcyVec_apply
import Mathlib
import Definitions.Def_ChapterNumericalRangeCrouzeix
open BookProof.ChapterNumericalRangeCrouzeix


open scoped InnerProductSpace


variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℂ E]

Formal statement

theorem BookProof.ChapterNumericalRangeCrouzeix.pearcyVec_apply (A : E →L[ℂ] E) (c : ℂ) (n : ℕ) (y : E) :
    pearcyVec A c n y - c • A (pearcyVec A c n y) = y - c ^ n • (A ^ n) y := by sorry

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