The Weierstrass Canonical Form of a Regular Matrix Pencil: Numerical Issues and Computational Techniques

2009 ◽  
pp. 322-329 ◽  
Author(s):  
Grigorios Kalogeropoulos ◽  
Marilena Mitrouli ◽  
Athanasios Pantelous ◽  
Dimitrios Triantafyllou
Keyword(s):  
Canonical Form ◽  
Matrix Pencil ◽  
Computational Techniques ◽  
Regular Matrix

scholarly journals On the Reduction of Singular Matrix Pencils

1935 ◽  
Vol 4 (2) ◽  
pp. 67-76 ◽  
Author(s):  
H. W. Turnbull
Keyword(s):  
Canonical Form ◽  
Matrix Pencil ◽  
Rational Method ◽  
Singular Matrix ◽  
Matrix Pencils ◽  
Minimal Index

The following rational method of dealing with the reduction of a singular matrix pencil to canonical form has certain advantages. It is based on the principle of vector chains, the length of the chain determining a minimal index. This treatment is analogous to that employed by Dr A. C. Aitken and the author in Canonical Matrices (1932) 45–57, for the nonsingular case. In Theorems 1 and 2 tests are explicitly given for determining the minimal indices. Theorem 2 gives a method of discovering the lowest row (or column) minimal index. Theoretically it should be possible to state a corresponding theorem for each of these indices, not necessarily the lowest, and prior to any reduction of the pencil. This extension still awaits solution.

scholarly journals On the Equivalence of Two Singular Matrix Pencils

1936 ◽  
Vol 4 (4) ◽  
pp. 224-231 ◽  
Author(s):  
J. Williamson
Keyword(s):  
Canonical Form ◽  
Matrix Pencil ◽  
Rational Method ◽  
Singular Matrix ◽  
Matrix Pencils ◽  
Minimal Index

In a recent paper Turnbull, discussing a rational method for the reduction of a singular matrix pencil to canonical form, has shown how the lowest row, or column, minimal index may be determined directly without reducing the pencil to canonical form. It is the purpose of this note to show how all such indices may be determined, and at the same time to give conditions, somewhat simpler than the usual ones, for the equivalence of two matrix pencils.

ON THE CANONICAL FORM OF THE SINGULAR MATRIX PENCIL

1933 ◽  
Vol os-4 (1) ◽  
pp. 241-257 ◽  
Author(s):  
A. C. AITKEN
Keyword(s):  
Canonical Form ◽  
Matrix Pencil ◽  
Singular Matrix

On the Kronecker canonical form of a full row rank matrix pencil and applications

2019 ◽  
Author(s):  
Sophia D Karathanasi ◽  
Nicholas P Karampetakis
Keyword(s):  
Canonical Form ◽  
Matrix Pencil ◽  
Differential Algebraic Equations ◽  
Descriptor Systems ◽  
Algebraic Equations ◽  
Closed Form Solutions ◽  
Transformation Matrices ◽  
Rank Matrix ◽  
The Right ◽  
Kronecker Canonical Form

Abstract The Kronecker canonical form (KCF) of matrix pencils plays an important role in many fields such as systems control and differential–algebraic equations. In this article, we compute a finite and infinite Jordan chain and also a singular chain of vectors corresponding to a full row rank matrix pencil using an extended algorithm, first introduced by Jones (1999, Ph.D. Thesis, Department of Mathematics, Loughborough University of Technology, Loughborough, UK). The proposed method exploits these vectors forming the chains corresponding to the finite and infinite eigenvalues and to the right minimal indices of the pencil. This leads to the computation of two transformation matrices for obtaining under strict equivalence the KCF of the pencil. An application to the study of homogeneous linear rectangular descriptor systems is considered and closed form solutions are obtained in terms of these two transformation matrices. All the results are illustrated with an example.

A new bound for the distance to singularity of a regular matrix pencil

PAMM ◽  
2017 ◽  
Vol 17 (1) ◽  
pp. 863-864 ◽  
Author(s):  
Thomas Berger ◽  
Hannes Gernandt ◽  
Carsten Trunk ◽  
Henrik Winkler ◽  
Michał Wojtylak
Keyword(s):  
Matrix Pencil ◽  
Regular Matrix
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