Theorems · Theorem · operator theory

LinearMap.IsSymmetric.toMatrix_eigenvectorBasis

∀ {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),
  (LinearMap.toMatrix (hT.eigenvectorBasis hn).toBasis (hT.eigenvectorBasis hn).toBasis) T =
    Matrix.diagonal (RCLike.ofReal ∘ hT.eigenvalues hn)

Defined in: Mathlib.Analysis.InnerProductSpace.Spectrum
Cited by: 2 results in Mathlib
Foundations: Depth 258 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: RCLike NormedAddCommGroup InnerProductSpace FiniteDimensional

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Cites32

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

DFunLike.coestatement and proof · cited by 62,936
Realstatement · cited by 25,697
RingHom.idstatement and proof · cited by 18,349
NormedAddCommGroupstatement and proof · cited by 15,752
LinearMapstatement and proof · cited by 10,215
Matrixstatement · cited by 4,303
mul_oneproof · cited by 3,885
InnerProductSpacestatement and proof · cited by 3,523
LinearEquivstatement · cited by 3,317
RCLikestatement and proof · cited by 2,829
FiniteDimensionalstatement and proof · cited by 1,854
Module.finrankstatement and proof · cited by 1,770

Cited by2

Results whose statement or proof uses this declaration.

LinearMap.IsSymmetric.roots_charpoly_eq_eigenvaluesproof · cited by 4
LinearMap.IsSymmetric.charpoly_eqproof · cited by 1