Least Squares Problems in Orthornormalization

الأصل · EN

For any n-tuple (α₁,...,αₙ) of linearly independent vectors in Hilbert space H, we construct a unique orthonormal basis (ε₁,...,εₙ) of span{α₁,...,αₙ} satisfying: ∑ᵢ₌₁ⁿεᵢ-αᵢ²≤∑ᵢ₌₁ⁿβᵢ-αᵢ² for all orthonormal basis (β₁,...,βₙ) of span{α₁,...,αₙ}. We study the stability of the orthornormalization and give some applications and examples.