A Simple Algorithm for Constructing all Real Hessenberg Unitary Matrices

TL;DR

This paper introduces a simple algorithm to construct all real Hessenberg unitary matrices, providing a systematic way to generate sparse matrices with customizable properties for various applications.

Contribution

The paper presents the first algorithm capable of constructing all real Hessenberg unitary matrices with adjustable parameters for specific application needs.

Findings

Algorithm efficiently generates all real Hessenberg unitary matrices.

Matrices have $n-1$ variables for customization.

Applicable to diverse analysis problems involving unitary matrices.

Abstract

Unitary matrices which are zero below the secondary diagonal (Hessenberg unitary matrices) have many uses in analysis. Given a set of needed conditions on a unitary matrix, this algorithm will give the sparsest unitary matrix. We give an algorithm for constructing all real Hessenberg unitary matrices. The $n\times n$ unitary matrices given by the algorithm have $n-1$ variables which can be chosen to give additional properties needed for a particular application.