Transitivity in polarization optics and the diagonalization of quadratic forms

TL;DR

This paper explores the mathematical properties of Mueller matrices in polarization optics, demonstrating how quadratic forms associated with these matrices can be diagonalized, revealing insights into the polarization state of light.

Contribution

It introduces a method to diagonalize quadratic forms related to Mueller matrices, clarifying their structure for different polarization states.

Findings

Quadratic forms associated with Mueller matrices can be diagonalized.

Partially polarized light corresponds to four positive diagonal coefficients.

Completely polarized light corresponds to two zero diagonal coefficients.

Abstract

Any one measurement with polarized light makes it possible to fix the Mueller matrices of the Lorentz type with up to four arbitrary numeric parameters (x, u; z, w). These parameters are subject to the quadratic condition. It is demonstrated that the quadratic form can be diagonalized; in the case of partially polarized light four diagonal coefficients turn out to benon-zero and positive; in the case of completely polarized light two diagonal coefficients equal to zero.