Theorems · Theorem · linear algebra

LinearMap.aeval_self_charpoly

∀ {R : Type u} {M : Type v} [inst : CommRing R] [inst_1 : AddCommGroup M] [inst_2 : Module R M]
  [inst_3 : Module.Free R M] [inst_4 : Module.Finite R M] (f : M →ₗ[R] M), (Polynomial.aeval f) f.charpoly = 0

The Cayley-Hamilton Theorem, that the characteristic polynomial of a linear map, applied to the linear map itself, is zero. See Matrix.aeval_self_charpoly for the equivalent statement about matrices.

Defined in

Mathlib.LinearAlgebra.Charpoly.Basic

Cited by

6 results in Mathlib

Foundations

Depth 126 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingAddCommGroupModuleModule.FreeModule.Finite