Theorems · Theorem · functional analysis

LinearMap.normDet_sq

∀ {𝕜 : Type u_1} {U : Type u_2} {V : Type u_3} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup U]
  [inst_2 : InnerProductSpace 𝕜 U] [inst_3 : FiniteDimensional 𝕜 U] [inst_4 : NormedAddCommGroup V]
  [inst_5 : InnerProductSpace 𝕜 V] [inst_6 : FiniteDimensional 𝕜 V] (f : U →ₗ[𝕜] V),
  ↑(f.normDet ^ 2) = LinearMap.det (LinearMap.adjoint f ∘ₗ f)

The square of f.normDet equals the determinant of f.adjoint ∘ₗ f when the codomain is finite dimensional.

Defined in

Mathlib.Analysis.InnerProductSpace.NormDet

Cited by

1 results in Mathlib

Foundations

Depth 249 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupInnerProductSpaceFiniteDimensionalNormedAddCommGroupInnerProductSpaceFiniteDimensional