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

Maria İsabel Garcia-planas; Maria Dolors Magret

doi:10.32323/ujma.454890

EN

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

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

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Details

Primary Language

English

Subjects

Mathematical Sciences

Journal Section

Research Article

Authors

Maria İsabel Garcia-planas ^*
0000-0001-7418-7208
Spain

Maria Dolors Magret
Spain

Publication Date

March 20, 2019

Submission Date

August 22, 2018

Acceptance Date

February 2, 2019

Published in Issue

Year 2019 Volume: 2 Number: 1

DOI

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

IZ

https://izlik.org/JA93AM98US

APA

Garcia-planas, M. İ., & Magret, M. D. (2019). $D$-Stratification and Hierarchy Graphs of the Space of Order 2 and 3 Matrix Pencils. Universal Journal of Mathematics and Applications, 2(1), 14-23. https://doi.org/10.32323/ujma.454890

AMA

1.Garcia-planas Mİ, Magret MD. $D$-Stratification and Hierarchy Graphs of the Space of Order 2 and 3 Matrix Pencils. Univ. J. Math. Appl. 2019;2(1):14-23. doi:10.32323/ujma.454890

Chicago

Garcia-planas, Maria İsabel, and Maria Dolors Magret. 2019. “$D$-Stratification and Hierarchy Graphs of the Space of Order 2 and 3 Matrix Pencils”. Universal Journal of Mathematics and Applications 2 (1): 14-23. https://doi.org/10.32323/ujma.454890.

EndNote

Garcia-planas Mİ, Magret MD (March 1, 2019) $D$-Stratification and Hierarchy Graphs of the Space of Order 2 and 3 Matrix Pencils. Universal Journal of Mathematics and Applications 2 1 14–23.

IEEE

[1]M. İ. Garcia-planas and M. D. Magret, “$D$-Stratification and Hierarchy Graphs of the Space of Order 2 and 3 Matrix Pencils”, Univ. J. Math. Appl., vol. 2, no. 1, pp. 14–23, Mar. 2019, doi: 10.32323/ujma.454890.

ISNAD

Garcia-planas, Maria İsabel - Magret, Maria Dolors. “$D$-Stratification and Hierarchy Graphs of the Space of Order 2 and 3 Matrix Pencils”. Universal Journal of Mathematics and Applications 2/1 (March 1, 2019): 14-23. https://doi.org/10.32323/ujma.454890.

JAMA

1.Garcia-planas Mİ, Magret MD. $D$-Stratification and Hierarchy Graphs of the Space of Order 2 and 3 Matrix Pencils. Univ. J. Math. Appl. 2019;2:14–23.

MLA

Garcia-planas, Maria İsabel, and Maria Dolors Magret. “$D$-Stratification and Hierarchy Graphs of the Space of Order 2 and 3 Matrix Pencils”. Universal Journal of Mathematics and Applications, vol. 2, no. 1, Mar. 2019, pp. 14-23, doi:10.32323/ujma.454890.

Vancouver

1.Maria İsabel Garcia-planas, Maria Dolors Magret. $D$-Stratification and Hierarchy Graphs of the Space of Order 2 and 3 Matrix Pencils. Univ. J. Math. Appl. 2019 Mar. 1;2(1):14-23. doi:10.32323/ujma.454890