Parametrization of 3x3 unitary matrices based on polarization algebra

TL;DR

This paper introduces a novel parametrization of 3x3 unitary matrices using polarization algebra, representing them through orthonormal polarization states and transformations, with applications to matrix decomposition.

Contribution

It presents a new mathematical framework for parametrizing 3x3 unitary matrices based on polarization algebra and Jones vectors, linking polarization states to matrix structure.

Findings

Parametrization depends on six independent parameters.

Representation as an orthogonal similarity transformation.

Application to the structure of Hermitian matrix decomposition.

Abstract

A parametrization of 3x3 unitary matrices is presented. This mathematical approach is inspired on polarization algebra and is formulated through the identification of a set of three orthonormal three-dimensional Jones vectors representing the respective pure polarization states. This approach leads to the representation of a 3x3 unitary matrix as an orthogonal similarity transformation of a particular type of unitary matrix that depends on six independent parameters, while the remaining three parameters correspond to the orthogonal matrix of the said transformation. The results obtained are applied to determine the structure of the second component of the characteristic decomposition of a 3x3 positive semidefinite Hermitian matrix.