Eigenvalue problem for an irregular $\lambda$-matrix

The solution of the eigenvalue problem is examined for the polynomial $D(\lambda)=A_0\lambda^2+A_1\lambda+A_2$ when the matrices $A_0$ and $A_2$ (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 matrix $D(\lambda)$ and to the zero eigenvalue of matrix $A_0$. The computation of the other eigenvalues of $D(\lambda)$ is reduced to the same problem for a constant matrix of lower dimension. An ALGOL program and test examples are presented.