The Jordan normal form; decomposition theorem for modules

1964 ◽  
Vol 15 (1) ◽  
pp. 276-281 ◽  
Author(s):  
J. L. Brenner
Keyword(s):  
Normal Form ◽  
Decomposition Theorem ◽  
Jordan Normal Form

scholarly journals Extension of matrix pencil reduction to abelian categories

2018 ◽  
Vol 17 (04) ◽  
pp. 1850062
Author(s):  
Olivier Verdier
Keyword(s):  
Normal Form ◽  
Vector Space ◽  
Decomposition Theorem ◽  
Abelian Category ◽  
Jordan Canonical Form ◽  
Main Technique ◽  
Abelian Categories ◽  
Category Of Modules ◽  
Space Case ◽  
And Control

Matrix pencils, or pairs of matrices, are used in a variety of applications. By the Kronecker decomposition theorem, they admit a normal form. This normal form consists of four parts, one part based on the Jordan canonical form, one part made of nilpotent matrices, and two other dual parts, which we call the observation and control part. The goal of this paper is to show that large portions of that decomposition are still valid for pairs of morphisms of modules or abelian groups, and more generally in any abelian category. In the vector space case, we recover the full Kronecker decomposition theorem. The main technique is that of reduction, which extends readily to the abelian category case. Reductions naturally arise in two flavors, which are dual to each other. There are a number of properties of those reductions which extend remarkably from the vector space case to abelian categories. First, both types of reduction commute. Second, at each step of the reduction, one can compute three sequences of invariant spaces (objects in the category), which generalize the Kronecker decomposition into nilpotent, observation and control blocks. These sequences indicate whether the system is reduced in one direction or the other. In the category of modules, there is also a relation between these sequences and the resolvent set of the pair of morphisms, which generalizes the regular pencil theorem. We also indicate how this allows to define invariant subspaces in the vector space case, and study the notion of strangeness as an example.

Algebraic properties of a digraph and its line digraph

2003 ◽  
Vol 04 (04) ◽  
pp. 377-393 ◽  
Author(s):  
C. Balbuena ◽  
D. Ferrero ◽  
X. Marcote ◽  
I. Pelayo
Keyword(s):  
Normal Form ◽  
Characteristic Polynomials ◽  
Jordan Normal Form ◽  
Line Digraph ◽  
Algebraic Properties ◽  
Adjacency Matrices ◽  
Number Of Cycles

Let G be a digraph, LG its line digraph and A(G) and A(LG) their adjacency matrices. We present relations between the Jordan Normal Form of these two matrices. In addition, we study the spectra of those matrices and obtain a relationship between their characteristic polynomials that allows us to relate properties of G and LG, specifically the number of cycles of a given length.

scholarly journals The geometry of modified Riemannian extensions

2009 ◽  
Vol 465 (2107) ◽  
pp. 2023-2040 ◽  
Author(s):  
E. Calviño-Louzao ◽  
E. García-Río ◽  
P. Gilkey ◽  
R. Vázquez-Lorenzo
Keyword(s):  
Normal Form ◽  
Space Form ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Jordan Normal Form ◽  
Jacobi Operators ◽  
Necessary And Sufficient

We show that every paracomplex space form is locally isometric to a modified Riemannian extension and gives necessary and sufficient conditions for a modified Riemannian extension to be Einstein. We exhibit Riemannian extension Osserman manifolds of signature (3, 3), whose Jacobi operators have non-trivial Jordan normal form and which are not nilpotent. We present new four-dimensional results in Osserman geometry.

scholarly journals Determining the structure of the Jordan normal form of a matrix by symbolic computation

1997 ◽  
Vol 252 (1-3) ◽  
pp. 221-259 ◽  
Author(s):  
T.Y. Li ◽  
Zhinan Zhang ◽  
Tianjun Wang
Keyword(s):  
Normal Form ◽  
Symbolic Computation ◽  
Jordan Normal Form
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