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

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.

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 Nearest Singular Matrix Pencil

2017 ◽  
Vol 38 (3) ◽  
pp. 776-806 ◽  
Author(s):  
Nicola Guglielmi ◽  
Christian Lubich ◽  
Volker Mehrmann
Keyword(s):  
Matrix Pencil ◽  
Singular Matrix

THE AUTOMORPHIC TRANSFORMATIONS OF A SINGULAR MATRIX PENCIL

1936 ◽  
Vol os-7 (1) ◽  
pp. 277-289
Author(s):  
W. LEDERMANN
Keyword(s):  
Matrix Pencil ◽  
Singular Matrix

scholarly journals On generalized inverses of singular matrix pencils

2011 ◽  
Vol 21 (1) ◽  
pp. 161-172 ◽  
Author(s):  
Klaus Röbenack ◽  
Kurt Reinschke
Keyword(s):  
Linear Time ◽  
Matrix Pencil ◽  
Differential Algebraic Equations ◽  
Generalized Inverses ◽  
Drazin Inverse ◽  
Algebraic Equations ◽  
Singular Matrix ◽  
Linear Time Invariant ◽  
Matrix Pencils ◽  
Linear Differential

On generalized inverses of singular matrix pencilsLinear time-invariant networks are modelled by linear differential-algebraic equations with constant coefficients. These equations can be represented by a matrix pencil. Many publications on this subject are restricted to regular matrix pencils. In particular, the influence of the Weierstrass structure of a regular pencil on the poles of its inverse is well known. In this paper we investigate singular matrix pencils. The relations between the Kronecker structure of a singular matrix pencil and the multiplicity of poles at zero of the Moore-Penrose inverse and the Drazin inverse of the rational matrix are investigated. We present example networks whose circuit equations yield singular matrix pencils.

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