isRoot_of_hasEigenvalue — Mathlib

English

An endomorphism of a finite-dimensional space has finitely many eigenvalues.

Русский

Эндоморфизм конечномерного пространства имеет finitely many eigenvalues.

LaTeX

$$$$ \text{The set } \{\mu \mid f \text{ has eigenvalue } \mu\} \text{ is finite}. $$$$

Lean4

theorem isRoot_of_hasEigenvalue [NoZeroSMulDivisors R M] {f : End R M} {μ : R} (h : f.HasEigenvalue μ) :
    (minpoly R f).IsRoot μ := by
  rcases (Submodule.ne_bot_iff _).1 h with ⟨w, ⟨H, ne0⟩⟩
  refine Or.resolve_right (smul_eq_zero.1 ?_) ne0
  rw [← aeval_apply_of_hasEigenvector ⟨H, ne0⟩]
  simp

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