Analysis of singular matrix pencils

One considers the spectral problem for a singular pencil $D(\lambda)=A+\lambda B$ of matrices $A$ and $B$ ($A$ and $B$ are rectangular matrices or $\det D(\lambda)=0$). One represents an algorithm which allows us to find the reducing subspaces for $D(\lambda)$ and with their aid to reduce the dimension of the initial pencil, by isolating from it the zero block, the blocks corresponding to the right and left polynomial solutions of the equations $(A+\lambda B)x(\lambda)=0$ and $y(\lambda)(A+\lambda B)=0$, respectively, as well as the block corresponding to the regular kernel of the pencil $D(\lambda)$. The algorithm is based on the application of the normalized process which uses the numerically stable elementary orthogonal transformations (the matrices of plane rotations or reflections).