An introduction to the Equiangular algorithm

TL;DR

This paper introduces the Equiangular algorithm, a generalization of Gram-Schmidt, for constructing equiangular vectors with a specified angle, leading to new matrix decompositions and analysis of their properties.

Contribution

The paper presents a novel algorithm for generating equiangular vectors with a given angle, extending Gram-Schmidt, and explores associated matrix properties and canonical forms.

Findings

Derived a new matrix decomposition based on equiangular vectors.

Analyzed inverse and eigenvalue problems for matrices from the algorithm.

Established properties and canonical forms of the resulting matrices.

Abstract

In this paper a generalization of the Gram-Schmidt Algorithm is presented. Actually we provide an algorithm to construct a set of equiangular vectors with a given angle $\theta\in(0,\arccos(\frac{-1}{n-1}))$ using a set of input independent vectors in $\mathbb{R}^n$. Therefore a usual type of matrix decomposition is derived. Then we discuss some properties of matrices derived from the new algorithm. The inverse and eigenvalue problems of these matrices if there exist are studied. Also, we derive some canonical forms based on the algorithm.