Theorems · Theorem · linear algebra

Module.End.iInf_maxGenEigenspace_restrict_map_subtype_eq

∀ {ι : Type u_1} {R : Type u_2} {M : Type u_4} [inst : CommRing R] [inst_1 : AddCommGroup M] [inst_2 : Module R M]
  (f : ι → Module.End R M) {μ : ι → R} (i : ι)
  (h : ∀ (j : ι), Set.MapsTo ⇑(f j) ↑((f i).maxGenEigenspace (μ i)) ↑((f i).maxGenEigenspace (μ i))),
  Submodule.map ((f i).maxGenEigenspace (μ i)).subtype
      (⨅ j, (fun j => Module.End.maxGenEigenspace (LinearMap.restrict (f j) ⋯) (μ j)) j) =
    ⨅ j, (f j).maxGenEigenspace (μ j)

Given a family of endomorphisms i ↦ f i, a family of candidate eigenvalues i ↦ μ i, and a distinguished index i whose maximal generalised μ i-eigenspace is invariant w.r.t. every f j, taking simultaneous maximal generalised eigenspaces is unaffected by first restricting to the distinguished generalised μ i-eigenspace.

Defined in

Mathlib.LinearAlgebra.Eigenspace.Pi

Cited by

1 results in Mathlib

Foundations

Depth 76 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingAddCommGroupModule