Regularizing algorithm for mixed matrix pencils

P. Van Dooren (1979) constructed an algorithm for computing all singular summands of Kronecker’s canonical form of a matrix pencil. His algorithm uses only unitary transformations, which improves its numerical stability. We extend Van Dooren’s algorithm to square complex matrices with respect to consimilarity transformations A↦SAS¯−1$\begin{array}{}
\displaystyle
A \mapsto SA{\bar S^{ - 1}}
\end{array}$ and to pairs of m × n complex matrices with respect to transformations (A,B)↦(SAR,SAR¯)$\begin{array}{}
\displaystyle
(A,B) \mapsto (SAR,SB\bar R)
\end{array}$, in which S and R are nonsingular matrices.