Canonical form with respect to semiscalar equivalence for a matrix pencil with nonsingular first matrix

2012 ◽  
Vol 63 (8) ◽  
pp. 1314-1320 ◽  
Author(s):  
V. M. Prokip
Keyword(s):  
Canonical Form ◽  
Matrix Pencil

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.

scholarly journals Regularizing algorithm for mixed matrix pencils

2017 ◽  
Vol 2 (1) ◽  
pp. 123-130 ◽  
Author(s):  
Tetiana Klymchuk
Keyword(s):  
Canonical Form ◽  
Numerical Stability ◽  
Matrix Pencil ◽  
Mixed Matrix ◽  
Complex Matrices ◽  
Regularizing Algorithm ◽  
Unitary Transformations ◽  
Nonsingular Matrices ◽  
Matrix Pencils ◽  
Square Complex

AbstractP. Van Dooren (1979) constructed an algorithm for computing all singular summands of Kronecker’s canonical form of a matrix pencil. His algorithm uses only unitary transformations, which improves its numerical stability. We extend Van Dooren’s algorithm to square complex matrices with respect to consimilarity transformations $\begin{array}{} \displaystyle A \mapsto SA{\bar S^{ - 1}} \end{array}$ and to pairs of m × n complex matrices with respect to transformations $\begin{array}{} \displaystyle (A,B) \mapsto (SAR,SB\bar R) \end{array}$, in which S and R are nonsingular matrices.

Sign in / Sign up

Export Citation Format

Share Document