A Note on Gram-Schmidt's Algorithm for a General Angle

TL;DR

This paper introduces a linear algorithm to construct vectors with prescribed angles or distances in inner product spaces, extending Gram-Schmidt's method, and explores its asymptotic behavior in infinite-dimensional spaces.

Contribution

It presents a novel linear algorithm for constructing vectors with specified angles, generalizing Gram-Schmidt's orthogonalization for all feasible cases.

Findings

Algorithm works in all cases where prescribed angles are possible

Constructs vectors with the same span and prescribed pairwise angles

Proves an asymptotic property in infinite-dimensional spaces

Abstract

The Gram-Schmidt algorithm produces a pairwise orthogonal set from a linearly independent set of vectors in an inner product vector space V. We give a linear algorithm that constructs vectors with the same span and which have pairwise the same prescribed angle or distance, in all cases where this is possible. Finally, we prove an asymptotic property in the case of an infinite dimensional space V.