$D$-Stratification and Hierarchy Graphs of the Space of Order 2 and 3 Matrix Pencils

Year 2019, , 14 - 23, 20.03.2019

Maria İsabel Garcia-planas Maria Dolors Magret

https://doi.org/10.32323/ujma.454890

Abstract

Small changes in the entries of a matrix pencil may lead to important changes in its Kronecker normal form. Studies about the effect of small perturbations have been made when considering the stratification associated with the strict equivalence between matrix pencils. In this work, we consider a partition in the space of pairs of matrices associated to regular matrix pencils, which will be proved to be a finite stratification of the space of such matrix pencils, called D-stratification. Matrix pencils in the same strata are those having some prescribed Segre indices. We study the effect of perturbations which lead to changes in the Kronecker canonical form, preserving the order of the nilpotent part. Our goal is to determine which $D$-strata can be reached. In the cases where the order of the matrix pencils is 2 or 3, we obtain the corresponding hierarchy graphs, illustrating the $D$-strata that can be reached when applying some small perturbations.

Keywords

Equivalence relation, Kronecker normal form, Linear systems, Matrix pencils, Stratification

References

• [1] V. I. Arnold, On matrices depending on parameters, Russian Math. Surveys, 26(2), (1971), 29–43.
• [2] V. I. Arnold, Singularity Theory, Cambridge University Press, 1981.
• [3] H. den Boer, G. Ph. A. Thijsse, Semi-stability of sums of partial multiplicities under additive perturbation, Integral Equations Operator Theory, 3 (1980), 23–42.
• [4] A. S. Markus, E. E. Parilis, The change of the Jordan structure of a matrix under small perturbations, Linear Algebra and its Applications, 54, (1983), 139–152.
• [5] M. A. Beitia, I. de Hoyos, I. Zaballa, The change of similarity invariants under row perturbations: Generic cases, Linear Algebra Appl., 429 (2008), 482–496.
• [6] M. A. Beitia, I. de Hoyos, I. Zaballa, The change of similarity invariants under row perturbations, Linear Algebra Appl., 429(5–6) (2008), 1302–1333.
• [7] L. Klimenko, V.V. Sergeichuk, An informal introduction to perturbations of matrices determined up to similarity or congruence, Sao Paulo Journal of Mathematical Sciences, 8(1), (2014), 1–22.
• [8] A. Dmytryshyn, V. Futorny, B. Kagström, L. Klimenko, V.V. Sergeichuk, Change of the congruence canonical form of 2-by-2 and 3-by-3 matrices under perturbations and bundles of matrices under congruence, Linear Algebra Appl., 469 (2015), 305–334.
• [9] C. G. Gibson, Regularity of the Segre Stratification, Math. Proc. Camb. Phil. Soc. 80, (1976), 91–97.
• [10] M. D. Magret, Stratification of the space of matrices defining linear dynamical systems, Int. J. Complex Systems in Science, 6(1), (2016), 29–35.
• [11] M. D. Magret, Perturbation of matrices preserving the Segre characteristics of the non-nilpotent Jordan blocks, CEDYA 2017 - Communications Book, (2017), 429–432.
• [12] J. Clotet, M. D. Magret, Familias diferenciables de inversas de Drazin, UPC Commons, 2005.
• [13] M. I. Garcia-Planas, Estudio geom´etrico de familias diferenciables de parejas de matrices: deformaciones versales, cambios de base globales, estratificacion de Brunvsky. Ph D. Thesis, UPC, 1994.
• [14] M. I. Garcia-Planas, V.V. Sergeichuk, Generic families of matrix pencils and their bifurcation diagrams, Linear Algebra and its Applications 332, (2001), 165–179.
• [15] M. I. Garcia-Planas, Kronecker stratification of the space of quadruples of matrices, SIAM Journal on Matrix Analysis and Applications, 19(4), (1998), 872–885.
• [16] A. Pokrzywa, On perturbations and the equivalence orbit of a matrix pencil, Linear Algebra and its Applications, 82, (1986), 99–121.
• [17] A. Edelman, E. Elmroth, B. K°agstr¨om, A geometric approach to perturbation theory of matrices and matrix pencils. Part II: A stratification-enhanced staircase algorithm, SIAM J. Matrix Anal. Appl., 31(3) (1999), 667–699.
• [18] E. Elmroth, P. Johansson, B. Kagström, Computation and presentation of graphs displaying closure hierarchies of Jordan and Kronecker structures, Numer. Linear Algebra Appl., 8(6-7) (2001), 381–399.
• [19] V. Futorny, L. Klimenko, V.V. Sergeichuk, Change of the congruence canonical form of 2-by-2 matrices under perturbations, Electron. J. Linear Algebra, 27, (2014), 146–154.
• [20] A. Edelman, E. Elmroth, B. Kagström, A geometric approach to perturbation theory of matrices and matrix pencils, Part I: Versal deformations, SIAM J. Matrix Anal. Appl., 18(3) (1997), 653–692.
• [21] F. R. Gantmacher, The Theory of Matrices, Chelsea, New York (1959).
• [22] L. Dai. Singular Control Systems, Springer Verlag, New York, 1989.
• [23] M. I. Garcia-Planas, V.V. Sergeichuk, Simplest miniversal deformations of matrices, matrx pencils, and contragredient matrix pencils, Linear Algebra and its Applications, 302/303, (1999), 45–61.
• [24] J. E. Humphreys, Linear Algebraic Groups, Springer-Verlag, (1981).