Theorems · Theorem · linear algebra

LinearMap.polyCharpolyAux_map_aeval

∀ {R : Type u_1} {L : Type u_2} {M : Type u_3} {ι : Type u_5} {ιM : Type u_7} [inst : CommRing R]
  [inst_1 : AddCommGroup L] [inst_2 : Module R L] [inst_3 : AddCommGroup M] [inst_4 : Module R M]
  (φ : L →ₗ[R] Module.End R M) [inst_5 : Fintype ι] [inst_6 : Fintype ιM] [inst_7 : DecidableEq ι]
  [inst_8 : DecidableEq ιM] (b : Module.Basis ι R L) (bₘ : Module.Basis ιM R M) (A : Type u_8) [inst_9 : CommRing A]
  [inst_10 : Algebra R A] [inst_11 : Module.Finite A (TensorProduct R A M)]
  [inst_12 : Module.Free A (TensorProduct R A M)] (x : ι → A),
  Polynomial.map (MvPolynomial.aeval x).toRingHom (φ.polyCharpolyAux b bₘ) =
    ((LinearMap.tensorProduct R A M M ∘ₗ LinearMap.baseChange A φ)
        ((Algebra.TensorProduct.basis A b).repr.symm (Finsupp.equivFunOnFinite.symm x))).charpoly

Defined in

Mathlib.Algebra.Module.LinearMap.Polynomial

Cited by

1 results in Mathlib

Foundations

Depth 115 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingAddCommGroupModuleAddCommGroupModuleFintypeFintypeDecidableEqDecidableEqCommRingAlgebraModule.FiniteModule.Free