Computing a canonical form of a matrix pencil

Miloud Sadkane; Roger Sidje; Miloud Sadkane; Roger Sidje

doi:10.3934/math.2024531

AIMS Mathematics

2024, Volume 9, Issue 5: 10882-10892. doi: 10.3934/math.2024531

Previous Article Next Article

Research article Special Issues

Computing a canonical form of a matrix pencil

Miloud Sadkane ^{1
,
,},
Roger Sidje ²

1.
Univ Brest, CNRS - UMR 6205, LMBA, 6, Avenue Le Gorgeu. 29238 Brest Cedex 3, France
2.
The University of Alabama, Department of Mathematics, P.O. Box 870350, Tuscaloosa, AL 35487, USA

Received: 27 December 2023 Revised: 09 March 2024 Accepted: 13 March 2024 Published: 19 March 2024
MSC : 15A21, 15A22, 65F15

Using the spectral projection onto the deflating subspace of a regular matrix pencil corresponding to the eigenvalues inside a specified region of the complex plane, we proposed a new method to compute a corresponding canonical form. The study included a perturbation analysis of the method as well as examples to illustrate its numerical and theoretical merits.
- canonical form,
- matrix pencil,
- spectral projection
Citation: Miloud Sadkane, Roger Sidje. Computing a canonical form of a matrix pencil[J]. AIMS Mathematics, 2024, 9(5): 10882-10892. doi: 10.3934/math.2024531

Related Papers:

Abstract

Using the spectral projection onto the deflating subspace of a regular matrix pencil corresponding to the eigenvalues inside a specified region of the complex plane, we proposed a new method to compute a corresponding canonical form. The study included a perturbation analysis of the method as well as examples to illustrate its numerical and theoretical merits.

References

[1]	Z. Bai, J. Demmel, M. Gu, An inverse free parallel spectral divide and conquer algorithm for nonsymmetric eigenproblems, Numer. Math., 76 (1997), 279–308. https://doi.org/10.1007/s002110050264 doi: 10.1007/s002110050264
[2]	P. Benner, S. W. R. Werner, Model reduction of descriptor systems with the MORLAB toolbox, IFAC PapersOnLine, 51 (2018), 547–552. https://doi.org/10.1016/j.ifacol.2018.03.092 doi: 10.1016/j.ifacol.2018.03.092
[3]	K. A. Cliffe, T. J. Garratt, A. Spence, Eigenvalues of block matrices arising from problems in fluid mechanics, SIAM J. Matrix Anal. Appl., 15 (1994), 1310–1318. https://doi.org/10.1137/S0895479892233230 doi: 10.1137/S0895479892233230
[4]	S. K. Godunov, Modern aspects of linear algebra, translations of mathematical monographs, American Math. Soc., 175 (1998).
[5]	S. K. Godunov, Problem of the dichotomy of the spectrum of a matrix, Siberian. Math. J., 27 (1986), 649–660. https://doi.org/10.1007/BF00969193 doi: 10.1007/BF00969193
[6]	I. Gohberg, S. Goldberg, M. Kaashoek, Classes of linear operators, In: Operator Theory: Advances and Applications, 1 (1990).
[7]	G. H. Golub, C. F. Van Loan, Matrix Computations, $4^{th}$ edition, Baltimore: Johns Hopkins University Press, 2013.
[8]	G. Hechme, Y. Nechepurenko, M. Sadkane, Efficient methods for computing spectral projectors for linearized hydrodynamic equations, SIAM J. Sci. Comput., 31 (2008), 667–686. https://doi.org/10.1137/050648122 doi: 10.1137/050648122
[9]	B. Kågström, P. Poromaa, Computing eigenspaces with specified eigenvalues of a regular matrix pair $(A, B)$ and condition estimation: theory, algorithms and software, Numer. Algor., 12 (1996), 369–407. https://doi.org/10.1007/BF02142813 doi: 10.1007/BF02142813
[10]	A. N. Malyshev, Parallel algorithm for solving some spectral problems of linear algebra, Linear Algebra Appl., 188/189 (1993), 489–520. https://doi.org/10.1016/0024-3795(93)90477-6 doi: 10.1016/0024-3795(93)90477-6
[11]	R. März, Canonical projections for linear differential algebraic equations, Comput. Math. Appl., 31 (1996), 121–135. https://doi.org/10.1016/0898-1221(95)00224-3 doi: 10.1016/0898-1221(95)00224-3
[12]	M. Sadkane, A. Touhami, Algorithm 918: specdicho: A MATLAB program of spectral dichotomy of regular matrix pencil, ACM Trans. Math. Software, 38 (2012), 1–13. https://doi.org/10.1145/2168773.2168780 doi: 10.1145/2168773.2168780
[13]	M. Sadkane, R. B. Sidje, Efficient computation of the spectral projections of regular matrix pairs, J. Comput. Appl. Math., 298 (2016), 72–81. https://doi.org/10.1016/j.cam.2015.11.035 doi: 10.1016/j.cam.2015.11.035
[14]	G. W. Stewart, J. G. Sun, Matrix Perturbation Theory, San Diego: Academic Press, 1990.
[15]	L. N. Trefethen, M. Embree, Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators, Princeton: Princeton 26 University Press, 2005.

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)