The solution of the eigenvalue problem is examined for the polynomial matrixD(λ)=Aoλs+A1λs−1+...+As when the matricesA0 andA2 (or one of them) are singular. A normalized process is used for solving the problem, permitting the determination of linearly independent eigenvectors corresponding to the zero eigenvalue of matrixD(λ) and to the zero eigenvalue of matrixA0. The computation of the other eigenvalues ofD(λ) is reduced to the same problem for a constant matrix of lower dimension. An ALGOL program and test examples are presented.