Theorems · Theorem · operator theory

LinearMap.IsSymmetric.eigenvectorBasis_apply_self_apply

∀ {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} [inst_3 : FiniteDimensional 𝕜 E] {n : ℕ} (hT : T.IsSymmetric) (hn : Module.finrank 𝕜 E = n) (v : E)
  (i : Fin n),
  ((hT.eigenvectorBasis hn).repr (T v)).ofLp i = ↑(hT.eigenvalues hn i) * ((hT.eigenvectorBasis hn).repr v).ofLp i

Diagonalization theorem, spectral theorem; version 2: A self-adjoint operator T on a finite-dimensional inner product space E acts diagonally on the identification of E with Euclidean space induced by an orthonormal basis of eigenvectors of T.

Defined in

Mathlib.Analysis.InnerProductSpace.Spectrum

Cited by

0 results in Mathlib

Foundations

Depth 258 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupInnerProductSpaceFiniteDimensional