Regularizing algorithm for mixed matrix pencils

Abstract

Abstract 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 ↦ SA S ¯ − 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 , SA R ¯ ) $\begin{array}{} \displaystyle (A,B) \mapsto (SAR,SB\bar R) \end{array}$ , in which S and R are nonsingular matrices.