Least Squares Problems in Orthornormalization

Abstract

For any $n$-tuple $({\alpha }_1,...,{\alpha }_n)$ of linearly independent vectors in Hilbert space $H$, we construct a unique orthonormal basis $({\epsilon }_1,...,{\epsilon }_n)$ of $span\{{\alpha }_1,...,{\alpha }_n\}$ satisfying: $$\sum _{i=1}^n\Vert {\epsilon }_i-{\alpha }_i{\Vert }^2\le \sum _{i=1}^n\Vert {\beta }_i-{\alpha }_i{\Vert }^2$$ for all orthonormal basis $({\beta }_1,...,{\beta }_n)$ of $span\{{\alpha }_1,...,{\alpha }_n\}$. We study the stability of the orthornormalization and give some applications and examples.

10 pages

References in corpus (1)

• Completeness of $n$--tuple of projections in $C^{\ast }$--algebras