Theorems · Theorem · linear algebra

Module.End.genEigenspace_restrict_eq_top

∀ {K : Type u_1} {V : Type u_2} [inst : Field K] [inst_1 : AddCommGroup V] [inst_2 : Module K V] {p : Submodule K V}
  {f : Module.End K V} [FiniteDimensional K V] {k : ℕ∞} (h : ∀ x ∈ p, f x ∈ p),
  ⨆ μ, (f.genEigenspace μ) k = ⊤ → ⨆ μ, (Module.End.genEigenspace (LinearMap.restrict f h) μ) k = ⊤

In finite dimensions, if the generalized eigenspaces of a linear endomorphism span the whole space then the same is true of its restriction to any invariant submodule.

Defined in

Mathlib.LinearAlgebra.Eigenspace.Triangularizable

Cited by

1 results in Mathlib

Foundations

Depth 124 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

FieldAddCommGroupModuleFiniteDimensional