Solution of the eigenvalue problem for the regular bundle $D(\lambda)=\lambda A_0-A_1$ using deflated subspaces

We consider the solution of the generalized eigenvalue problem
$$ (A_0\lambda-A_1)x=0, $$
in the case where one or both of the matrices $A_0$, $A_1$ are degenerate but the intersection of their null spaces is empty. Using orthogonal matrices $\mathscr P$ and which are independent of $\lambda$ the original problem is transformed to a simpler one in which the pencil is of smaller dimension. The construction of $P$ and $Q$ uses the normalization process. We include an Algol program and sample runs.