Diagonal Scalings for the Eigenstructure of Arbitrary Pencils

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Journal Title
Journal ISSN
Volume Title
A1 Alkuperäisartikkeli tieteellisessä aikakauslehdessä
Date
2022
Major/Subject
Mcode
Degree programme
Language
en
Pages
25
1213-1237
Series
SIAM Journal on Matrix Analysis and Applications, Volume 43, issue 3
Abstract
In this paper we show how to construct diagonal scalings for arbitrary matrix pencils λB − A, in which both A and B are complex matrices (square or nonsquare). The goal of such diagonal scalings is to “balance” in some sense the row and column norms of the pencil. We see that the problem of scaling a matrix pencil is equivalent to the problem of scaling the row and column sums of a particular nonnegative matrix. However, it is known that there exist square and nonsquare nonnegative matrices that cannot be scaled arbitrarily. To address this issue, we consider an approximate embedded problem, in which the corresponding nonnegative matrix is square and can always be scaled. The new scaling methods are then based on the Sinkhorn–Knopp algorithm for scaling a square nonnegative matrix with total support to be doubly stochastic or on a variant of it. In addition, using results of U. G. Rothblum and H. Schneider [Linear Algebra Appl., 114–115 (1989), pp. 737–764], we give simple sufficient conditions on the zero pattern for the existence of diagonal scalings of square nonnegative matrices to have any prescribed common vector for the row and column sums. We illustrate numerically that the new scaling techniques for pencils improve the accuracy of the computation of their eigenvalues.
Description
Funding Information: This publication is part of the “Proyecto de I+D+i PID2019-106362GB-I00 financiado por MCIN/AEI/10.13039/501100011033.” The work of the authors was supported by “Ministerio de Economía, Industria y Competitividad (MINECO)” of Spain and “Fondo Europeo de Desarrollo Regional (FEDER)” of the EU through grants MTM2015-65798-P and MTM2017-90682-REDT and by the Madrid Government (Comunidad de Madrid-Spain) under the Multiannual Agreement with UC3M in the line of Excellence of University Professors (EPUC3M23) in the V PRICIT (Regional Programme of Research and Technological Innovation). The work of the second author was supported by the “contrato predoctoral” BES-2016-076744 of MINECO and by an Academy of Finland grant (Suomen Akatemian päätös 331240). This work was partially developed while the third author held a “Chair of Excellence UC3M - Banco de Santander” at Universidad Carlos III de Madrid in the academic year 2019–2020. The authors sincerely thank two anonymous referees for pointing out several significant suggestions and a number of relevant references that have contributed to improving this manuscript. Funding Information: ∗Received by the editors September 2, 2020; accepted for publication (in revised form) May 5, 2022; published electronically July 18, 2022. https://doi.org/10.1137/20M1364011 Funding: This publication is part of the “Proyecto de I+D+i PID2019-106362GB-I00 financiado por MCIN/AEI/10.13039/501100011033.” The work of the authors was supported by “Ministerio de Economía, Industria y Competitividad (MINECO)” of Spain and “Fondo Europeo de Desarrollo Regional (FEDER)” of the EU through grants MTM2015-65798-P and MTM2017-90682-REDT and by the Madrid Government (Comunidad de Madrid-Spain) under the Multiannual Agreement with UC3M in the line of Excellence of University Professors (EPUC3M23) in the V PRICIT (Regional Programme of Research and Technological Innovation). The work of the second author was supported by the “contrato predoctoral” BES-2016-076744 of MINECO and byan Academy of Finland grant (Suomen Akatemian päätös 331240). This work was partially developed while the third author held a “Chair of Excellence UC3M - Banco de Santander” at Universidad Carlos III de Madrid in the academic year 2019–2020. Publisher Copyright: © 2022 Society for Industrial and Applied Mathematics.
Keywords
accuracy of computed eigenvalues, diagonal scaling, pencils, Sinkhorn–Knopp algorithm
Other note
Citation
Dopico, F M, Quintana, M C & van Dooren, P 2022, ' Diagonal Scalings for the Eigenstructure of Arbitrary Pencils ', SIAM Journal on Matrix Analysis and Applications, vol. 43, no. 3, pp. 1213-1237 . https://doi.org/10.1137/20M1364011