Theorems · Theorem · operator theory

LinearMap.IsSymmetric.directSum_isInternal_of_pairwise_commute

∀ {𝕜 : Type u_1} {E : Type u_2} {n : Type u_3} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup E]
  [inst_2 : InnerProductSpace 𝕜 E] {T : n → Module.End 𝕜 E} [FiniteDimensional 𝕜 E] [inst_4 : DecidableEq (n → 𝕜)],
  (∀ (i : n), LinearMap.IsSymmetric (T i)) →
    Pairwise (Function.onFun Commute T) → DirectSum.IsInternal fun α => ⨅ j, (T j).eigenspace (α j)

In finite dimensions, given a commuting family of symmetric linear operators, the inner product space on which they act decomposes as an internal direct sum of joint eigenspaces.

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

RCLikeNormedAddCommGroupInnerProductSpaceFiniteDimensionalDecidableEq