lowerInv_eq_gramSchmidtBasis — Mathlib

English

The Kronecker product of a diagonal with a single entry results in a single diagonal entry at the corresponding paired position.

Русский

Кронекерово произведение диагонали и единичной матрицы даёт диагональ с соответствующим размещением.

LaTeX

$$diagonal a ⊗ₖₜ[R] single i j b = diagonal (i, j) (a_i ⊗ b)$$

Lean4

theorem lowerInv_eq_gramSchmidtBasis :
    LDL.lowerInv hS =
      ((Pi.basisFun 𝕜 n).toMatrix
          (@gramSchmidtBasis 𝕜 (n → 𝕜) _ (_ :) (InnerProductSpace.ofMatrix hS.transpose) n _ _ _ (Pi.basisFun 𝕜 n)))ᵀ :=
  by
  letI := NormedAddCommGroup.ofMatrix hS.transpose
  letI := InnerProductSpace.ofMatrix hS.transpose
  ext i j
  rw [LDL.lowerInv, Basis.coePiBasisFun.toMatrix_eq_transpose, coe_gramSchmidtBasis]
  rfl

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