Theorems · Theorem · functional analysis

InnerProductSpace.gramSchmidtOrthonormalBasis_inv_isUpperTriangular

∀ {𝕜 : Type u_1} {E : Type u_2} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace 𝕜 E]
  {ι : Type u_3} [inst_3 : LinearOrder ι] [inst_4 : LocallyFiniteOrderBot ι] [inst_5 : WellFoundedLT ι]
  [inst_6 : Fintype ι] [inst_7 : FiniteDimensional 𝕜 E] (h : Module.finrank 𝕜 E = Fintype.card ι) (f : ι → E),
  ((InnerProductSpace.gramSchmidtOrthonormalBasis h f).toBasis.toMatrix f).IsUpperTriangular

Given an indexed family f : ι → E of vectors in an inner product space E, for which the size of the index set is the dimension of E, the matrix of coefficients of f with respect to the orthonormal basis gramSchmidtOrthonormalBasis constructed from f is upper-triangular.

Defined in

Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho

Cited by

2 results in Mathlib

Foundations

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

Assumes

RCLikeNormedAddCommGroupInnerProductSpaceLinearOrderLocallyFiniteOrderBotWellFoundedLTFintypeFiniteDimensional