Theorems · Theorem · operator theory

LinearMap.IsSymmetric.iSup_iSup_eigenspace_inf_eigenspace_eq_top_of_commute

∀ {𝕜 : Type u_1} {E : Type u_2} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace 𝕜 E]
  {A B : E →ₗ[𝕜] E} [FiniteDimensional 𝕜 E],
  A.IsSymmetric → B.IsSymmetric → Commute A B → ⨆ α, ⨆ γ, Module.End.eigenspace A α ⊓ Module.End.eigenspace B γ = ⊤

If A and B are commuting symmetric operators acting on a finite-dimensional inner product space, then the simultaneous eigenspaces of A and B exhaust the space.

Defined in

Mathlib.Analysis.InnerProductSpace.JointEigenspace

Cited by

1 results in Mathlib

Foundations

Depth 205 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupInnerProductSpaceFiniteDimensional