Theorems · Theorem · operator theory

LinearMap.IsSymmetric.invariant_orthogonalComplement_eigenspace

∀ {𝕜 : Type u_1} [inst : RCLike 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace 𝕜 E]
  {T : E →ₗ[𝕜] E}, T.IsSymmetric → ∀ (μ : 𝕜), ∀ v ∈ (Module.End.eigenspace T μ)ᗮ, T v ∈ (Module.End.eigenspace T μ)ᗮ

A self-adjoint operator preserves orthogonal complements of its eigenspaces.

Defined in

Mathlib.Analysis.InnerProductSpace.Spectrum

Cited by

1 results in Mathlib

Foundations

Depth 162 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupInnerProductSpace