Limit Jordan normal form of large triangular matrices over a finite field

1996 ◽  
Vol 29 (4) ◽  
pp. 279-281 ◽  
Author(s):  
A. M. Borodin
Keyword(s):  
Normal Form ◽  
Finite Field ◽  
Jordan Normal Form ◽  
Triangular Matrices

FAST PARALLEL COMPUTATION OF THE JORDAN NORMAL FORM OF MATRICES

1996 ◽  
Vol 06 (02) ◽  
pp. 203-212 ◽  
Author(s):  
JEAN-LOUIS ROCH ◽  
GILLES VILLARD
Keyword(s):  
Normal Form ◽  
Finite Field ◽  
Parallel Computation ◽  
Characteristic Zero ◽  
Jordan Normal Form

We establish that the problem of computing the Jordan normal form of a matrix over a field F is in [Formula: see text] for F being a field of characteristic zero or a finite field.

scholarly journals Common Factors in Fraction-Free Matrix Decompositions

2020 ◽  
Author(s):  
Johannes Middeke ◽  
David J. Jeffrey ◽  
Christoph Koutschan
Keyword(s):  
Normal Form ◽  
Common Factors ◽  
Reduction Process ◽  
Triangular Matrices ◽  
Specific Data ◽  
Matrix Decompositions ◽  
The Matrix ◽  
The Common

AbstractWe consider LU and QR matrix decompositions using exact computations. We show that fraction-free Gauß–Bareiss reduction leads to triangular matrices having a non-trivial number of common row factors. We identify two types of common factors: systematic and statistical. Systematic factors depend on the reduction process, independent of the data, while statistical factors depend on the specific data. We relate the existence of row factors in the LU decomposition to factors appearing in the Smith–Jacobson normal form of the matrix. For statistical factors, we identify some of the mechanisms that create them and give estimates of the frequency of their occurrence. Similar observations apply to the common factors in a fraction-free QR decomposition. Our conclusions are tested experimentally.

Automorphisms of the total graph over upper triangular matrices

2019 ◽  
Vol 19 (08) ◽  
pp. 2050161
Author(s):  
Long Wang ◽  
Xianwen Fang ◽  
Fenglei Tian
Keyword(s):  
Finite Field ◽  
Total Graph ◽  
Triangular Matrices ◽  
Upper Triangular Matrices ◽  
Zero Divisors ◽  
Singular Matrices

Let [Formula: see text] be a finite field, [Formula: see text] the ring of all [Formula: see text] upper triangular matrices over [Formula: see text], [Formula: see text] the set of all zero-divisors of [Formula: see text], i.e. [Formula: see text] consists of all [Formula: see text] upper triangular singular matrices over [Formula: see text]. The total graph of [Formula: see text], denoted by [Formula: see text], is a graph with all elements of [Formula: see text] as vertices, and two distinct vertices [Formula: see text] are adjacent if and only if [Formula: see text]. In this paper, we determine all automorphisms of the total graph [Formula: see text] of [Formula: see text].

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

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.

Smith Normal Form of Augmented Degree Matrix and Rational Points on Toric Hypersurface

2013 ◽  
Vol 20 (02) ◽  
pp. 327-332
Author(s):  
Jianming Chen ◽  
Wei Cao
Keyword(s):  
Normal Form ◽  
Finite Field ◽  
Rational Points ◽  
Smith Normal Form ◽  
Degree Matrix

We use the Smith normal form of the augmented degree matrix to estimate the number of rational points on a toric hypersurface over a finite field. This is the continuation of a previous work by Cao in 2009.

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.

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