Theorems · Theorem · operator theory

LinearMap.IsSymmetric.roots_charpoly_eq_eigenvalues

∀ {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),
  T.charpoly.roots = Multiset.map (RCLike.ofReal ∘ hT.eigenvalues hn) Finset.univ.val

Defined in: Mathlib.Analysis.InnerProductSpace.Spectrum
Cited by: 4 results in Mathlib
Foundations: Depth 259 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: RCLike NormedAddCommGroup InnerProductSpace FiniteDimensional

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Cites36

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

DFunLike.coeproof · 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
Polynomialproof · cited by 5,681
Matrixproof · cited by 4,303
InnerProductSpacestatement and proof · cited by 3,523
Finset.univstatement and proof · cited by 3,473
RCLikestatement and proof · cited by 2,829
Multisetstatement and proof · cited by 2,627
FiniteDimensionalstatement and proof · cited by 1,854

Cited by4

Results whose statement or proof uses this declaration.

LinearMap.IsSymmetric.splits_charpolyproof · cited by 2
LinearMap.IsSymmetric.sort_roots_charpoly_eq_eigenvaluesproof · cited by 1
LinearMap.IsSymmetric.det_eq_prod_eigenvaluesproof · cited by 1
LinearMap.IsSymmetric.trace_eq_sum_eigenvaluesproof · cited by 1