Recursive parameterisation and invariant phases of unitary matrices

TL;DR

This paper explores a recursive method for parameterising n-by-n unitary matrices, allowing flexible factor ordering, and investigates invariant phases and symmetric matrices, with detailed analysis for 4x4 cases.

Contribution

It introduces a flexible recursive parameterisation scheme for unitary matrices and applies it to study invariant phases and symmetric matrices.

Findings

Factors can be introduced in any order in the recursive scheme.

Detailed analysis of 4x4 unitary matrices.

Method for constructing symmetric unitary matrices.

Abstract

We present further properties of a previously proposed recursive scheme for parameterisation of n-by-n unitary matrices. We show that the factors in the recursive formula may be introduced in any desired order. The method is used to study the invariant phases of unitary matrices. The case of four-by-four unitary matrices is investigated in detail. We also address the question of how to construct symmetric unitary matrices using the recursive approach.