The Linearly Independent Non Orthogonal yet Energy Preserving (LINOEP) vectors

TL;DR

This paper introduces a novel transformation method called LINOEP that converts linearly independent vectors into non-orthogonal vectors that preserve energy, expanding the possibilities beyond traditional orthogonalization.

Contribution

The paper proposes the LINOEP method, a new transformation that produces energy-preserving, non-orthogonal, linearly independent vectors in an inner product space, differing from standard orthogonalization techniques.

Findings

LINOEP preserves the energy (norm squared) of vectors.

Multiple solutions exist for energy preservation.

LINOEP extends the concept of vector transformation beyond orthogonalization.

Abstract

It is well known that, in any inner product space, a set of linearly independent (LI) vectors can be transformed to a set of orthogonal vectors, spanning the same space, by the Gram-Schmidt Orthogonalization Method (GSOM). In this paper, we propose a transformation from a set of LI vectors to a set of LI non orthogonal yet energy (square of the norm) preserving (LINOEP) vectors in an inner product space and we refer it as LINOEP method. We also show that there are various solutions to preserve the square of the norm.