A. Proof of the Jordan normal form

2017 ◽  
Keyword(s):  
Normal Form ◽  
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.

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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