Theorems · Theorem · linear algebra

LinearMap.polyCharpolyAux_basisIndep

∀ {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) {ιM' : Type u_8} [inst_9 : Fintype ιM']
  [inst_10 : DecidableEq ιM'] (bₘ' : Module.Basis ιM' R M), φ.polyCharpolyAux b bₘ = φ.polyCharpolyAux b bₘ'

LinearMap.polyCharpolyAux is independent of the choice of basis of the target module. Proof strategy: 1. Rewrite polyCharpolyAux as the (honest, ordinary) characteristic polynomial of the base change of φ to the multivariate polynomial ring MvPolynomial ι R. 2. Use that the characteristic polynomial of a linear map is independent of the choice of basis. This independence result is used transitively via LinearMap.polyCharpolyAux_map_aeval and LinearMap.polyCharpolyAux_map_eq_charpoly.

Defined in

Mathlib.Algebra.Module.LinearMap.Polynomial

Cited by

2 results in Mathlib

Foundations

Depth 116 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingAddCommGroupModuleAddCommGroupModuleFintypeFintypeDecidableEqDecidableEqFintypeDecidableEq