Degenerate 4-Dimensional Matrices with Semi-Group Structure
  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 
O. Veko, E. Ovsiyuk, A. Oana, M. Neagu, V. Balan, V. Red'kov
math.GM physics.optics