<abstract xmlns="http://www.w3.org/1999/xhtml"><p>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 <inline-formula><alternatives><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mi>A</mi><mo>↦</mo><msup><mrow><mtext>SA</mtext><mover><mrow><mi>S</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></math><tex-math>$\begin{array}{}
\displaystyle
A \mapsto SA{\bar S^{ - 1}}
\end{array}$</tex-math></alternatives></inline-formula> and to pairs of <italic>m</italic> × <italic>n</italic> complex matrices with respect to transformations <inline-formula><alternatives><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mo>(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>)</mo></mrow><mo>↦</mo><mrow><mo>(</mo><mtext>SAR</mtext><mo>,</mo><mtext>SA</mtext><mover><mrow><mi>R</mi></mrow><mrow><mo>¯</mo></mrow></mover><mo>)</mo></mrow></mrow></math><tex-math>$\begin{array}{}
\displaystyle
(A,B) \mapsto (SAR,SB\bar R)
\end{array}$</tex-math></alternatives></inline-formula>, in which <italic>S</italic> and <italic>R</italic> are nonsingular matrices.</p></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↦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.

Regularizing algorithm for mixed matrix pencils

Taras Shevchenko National University of Kyiv

Applied Mathematics and Nonlinear Sciences

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

{"article-title":"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...

Regularizing algorithm for mixed matrix pencils

Published Online: Apr 18, 2017

Page range: 123 - 130

Received: Feb 06, 2017

Accepted: Apr 18, 2017

DOI: https://doi.org/10.21042/AMNS.2017.1.00010

Keywords
Regularizing algorithm, Matrix pencils, Consimilarity, Unitary transformations, Canonical forms

© 2017 Tetiana Klymchuk, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Regularizing algorithm for mixed matrix pencils

Published Online: Apr 18, 2017

Page range: 123 - 130

Received: Feb 06, 2017

Accepted: Apr 18, 2017

DOI: https://doi.org/10.21042/AMNS.2017.1.00010

KeywordsRegularizing algorithm, Matrix pencils, Consimilarity, Unitary transformations, Canonical forms

© 2017 Tetiana Klymchuk, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
Regularizing algorithm, Matrix pencils, Consimilarity, Unitary transformations, Canonical forms