aeval_eq_aeval_mod_charpoly — Mathlib

English

If a linear map f is injective under certain compatibility conditions, a specific coefficient relation holds between minpoly and f.

Русский

Если линейное отображение f инъективно при заданной совместимости, выполняется равенство коэффициентов между minpoly и f.

LaTeX

$$minpoly_coeff_zero_of_injective ... (hf : Injective f) → (minpoly R f).coeff 0 ≠ 0$$

Lean4

/-- Any endomorphism polynomial `p` is equivalent under evaluation to `p %ₘ f.charpoly`; that is,
`p` is equivalent to a polynomial with degree less than the dimension of the module. -/
theorem aeval_eq_aeval_mod_charpoly (p : R[X]) : aeval f p = aeval f (p %ₘ f.charpoly) :=
  (aeval_modByMonic_eq_self_of_root f.charpoly_monic f.aeval_self_charpoly).symm

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