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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>A</m:mi> <m:mo>↦</m:mo> <m:msup> <m:mrow> <m:mtext>SA</m:mtext> <m:mover> <m:mrow> <m:mi>S</m:mi> </m:mrow> <m:mrow> <m:mo>¯</m:mo> </m:mrow> </m:mover> </m:mrow> <m:mrow> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> </m:mrow> </m:math> array A ↦ SA S- 1 array and to pairs of m × n complex matrices with respect to transformations <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mi>A</m:mi> <m:mo>,</m:mo> <m:mi>B</m:mi> <m:mo>)</m:mo> </m:mrow> <m:mo>↦</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mtext>SAR</m:mtext> <m:mo>,</m:mo> <m:mtext>SA</m:mtext> <m:mover> <m:mrow> <m:mi>R</m:mi> </m:mrow> <m:mrow> <m:mo>¯</m:mo> </m:mrow> </m:mover> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> array (A,B) ↦ (SAR,SB R) array , in which S and R are nonsingular matrices.