Degenerate 4-Dimensional Matrices with Semi-Group Structure

TL;DR

This paper classifies degenerate 4x4 real matrices with semi-group structures, focusing on matrices of ranks 1, 2, and 3, and explores their algebraic properties relevant to Mueller matrices.

Contribution

It introduces a parameterization technique for 4x4 matrices and classifies degenerate matrices into subgroups and semi-groups based on rank and linear restrictions.

Findings

Classified degenerate matrices of ranks 1, 2, and 3

Identified conditions for matrices to form subgroups or semi-groups

Specified 16 cases for rank 3 singular matrices

Abstract

While dealing with the nontrivial task of classifying Mueller matrices, of special interest is the study of the degenerate Mueller matrices (matrices with vanishing determinant, for which the law of multiplication holds, but there exists no inverse elements). Earlier, it was developed a special technique of parameterizing arbitrary 4-dimensional matrices with the use of a four 4-dimensional vector (k,m,l,n). In the following, a classification of degenerate 4-dimensional real matrices of rank 1, 2, and 3 is elaborated. To separate possible classes of degenerate matrices of ranks 1 and 2, we impose linear restrictions on (k,m, l,n), which are compatible with the group multiplication law. All the subsets of matrices obtained by this method, form either subgroups or semi-groups. To obtain singular matrices of rank 3, we specify 16 independent possibilities to get the 4-dimensional matrices with zero determinant