Theorems · Theorem · operator theory

LinearMap.IsSymmetric.det_eq_prod_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),
  LinearMap.det T = ∏ i, ↑(hT.eigenvalues hn i)

Defined in: Mathlib.Analysis.InnerProductSpace.Spectrum
Cited by: 1 results in Mathlib
Foundations: Depth 261 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: RCLike NormedAddCommGroup InnerProductSpace FiniteDimensional

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Cites23

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

DFunLike.coestatement · cited by 62,936
RingHom.idstatement and proof · cited by 18,349
NormedAddCommGroupstatement and proof · cited by 15,752
LinearMapstatement and proof · cited by 10,215
MonoidHomstatement · cited by 3,629
InnerProductSpacestatement and proof · cited by 3,523
Finset.univstatement and proof · cited by 3,473
RCLikestatement and proof · cited by 2,829
Finset.prodstatement and proof · cited by 2,356
FiniteDimensionalstatement and proof · cited by 1,854
Module.finrankstatement and proof · cited by 1,770
Multiset.prodproof · cited by 528

Cited by1

Results whose statement or proof uses this declaration.

LinearMap.normDet_eq_prod_singularValuesproof · cited by 0