  In polarization optics, an important role play Mueller matrices -- real
four-dimensional matrices which describe the effect of action of optical
elements on the polarization state of the light, described by 4-dimensional
Stokes vectors. An important issue is to classify possible classes of the
Mueller matrices. In particular, of special interest are degenerate Mueller
matrices with vanishing determinants. Earlier, it was developed a special
technique of parameterizing arbitrary 4-dimensional matrices with the use of
four 4-dimensional vector (k, m, l, n). In the paper, a classification of
degenerate 4-dimensional real matrices of rank 1, 2, 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, are
either sub-groups or semigroups. To obtain singular matrices of rank 3, we
specify 16 independent possibilities to get 4-dimensional matrices with zero
determinant.
