Theorems · Definition · functional analysis

LinearMap.normDet

{𝕜 : Type u_1} →
  {U : Type u_2} →
    {V : Type u_3} →
      [inst : RCLike 𝕜] →
        [inst_1 : NormedAddCommGroup U] →
          [inst_2 : InnerProductSpace 𝕜 U] →
            [FiniteDimensional 𝕜 U] →
              [inst_4 : NormedAddCommGroup V] → [inst_5 : InnerProductSpace 𝕜 V] → (U →ₗ[𝕜] V) → ℝ

The norm determinant of a linear map f : U →ₗ[𝕜] V is defined as the norm of the determinant of the square matrix representing the linear map U →ₗ[𝕜] f.range over a pair of orthonormal basis of equal dimensions. (See LinearMap.normDet_eq_norm_det_toMatrix_rangeRestrict for using arbitrary orthonormal basis) If such basis doesn't exist (e.g. the map is not injective), the norm determinant is zero. (See LinearMap.normDet_eq_zero_iff_ker_ne_bot)

Defined in

Mathlib.Analysis.InnerProductSpace.NormDet

Cited by

27 results in Mathlib

Foundations

Depth 241 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupInnerProductSpaceFiniteDimensionalNormedAddCommGroupInnerProductSpace