Diagonalization over Commutative Rings

Richter, R. Bruce; Wardlaw, William P.

doi:10.1080/00029890.1990.11995580

Cited by 3 publications

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“…But even in the special case when G = Z n , it is not always true that λ is then a root in G of the equation det(L − λI ) = 0. However, the implication Lā = λā ⇒ det(L − λI ) = 0 does hold in the important special case when the entries ofā form a generating set for the group Z n [16].…”

Section: Theorem 5 Let Be a Connected Graph Let T Be A Spanning Tree Of And Let α Be A Proper T -Reduced Voltage Assignment On In An Abelmentioning

confidence: 99%

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Širáň‬

2001

Journal of Algebraic Combinatorics

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Lifts of graph and map automorphisms can be described in terms of voltage assignments that are, in a sense, compatible with the automorphisms. We show that compatibility of ordinary voltage assignments in Abelian groups is related to orthogonality in certain Z-modules. For cyclic groups, compatibility turns out to be equivalent with the existence of eigenvectors of certain matrices that are naturally associated with graph automorphisms. This allows for a great simplification in characterizing compatible voltage assignments and has applications in constructions of highly symmetric graphs and maps.

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Section: Theorem 5 Let Be a Connected Graph Let T Be A Spanning Tree Of And Let α Be A Proper T -Reduced Voltage Assignment On In An Abelmentioning

confidence: 99%

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Širáň‬

2001

Journal of Algebraic Combinatorics

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“…Diagonalization and finding eigenvalues and eigenvectors of a matrix defined on a ring is not as easy as in matrices defined over a field. There are many studies on this subject in the literature [18], [21], [15]. In particular, exponential of a matrix defined on the ring of quaternions can be found in Casey's paper [17].…”

Section: Introductionmentioning

confidence: 99%

Explicit formulas for exponential of 2×2 split-complex matrices

Çakır

Özdemir

2022

Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat.

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Split-complex (hyperbolic) numbers are ordered pairs of real numbers, written in the form $x+jy$ with $j^{2}=-1$, used to describe the geometry of the Lorentzian plane. Since a null split-complex number does not have an inverse, some methods to calculate the exponential of complex matrices are not valid for split-complex matrices. In this paper, we examined the exponential of a $2x2$ split-complex matrix in three cases : $i:~\Delta=0,~ii:~\Delta\neq0$ and $\Delta$ is not null split-complex number, $iii:~\Delta\neq0$ and $\Delta$ is a null split-complex number where $\Delta=(trA)^{2}-4detA$.

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Linear dynamical systems of dimension two over the ring of integers modulo pt

Wei

Liang

2020

Discrete Math. Algorithm. Appl.

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In this paper, we investigate the linear dynamical system [Formula: see text], where [Formula: see text] is the ring of integers modulo [Formula: see text] ([Formula: see text] is a prime). In order to facilitate the visualization of this system, we associate a graph [Formula: see text] on it, whose nodes are the points of [Formula: see text], and for which there is an arrow from [Formula: see text] to [Formula: see text], when [Formula: see text] for a fixed [Formula: see text] matrix [Formula: see text]. In this paper, the in-degree of each node in [Formula: see text] is obtained, and a complete description of [Formula: see text] is given, when [Formula: see text] is an idempotent matrix, or a nilpotent matrix, or a diagonal matrix. The results in this paper generalize Elspas’ [1959] and Toledo’s [2005].

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Diagonalization over Commutative Rings

Cited by 3 publications

References 2 publications

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Explicit formulas for exponential of 2×2 split-complex matrices

Linear dynamical systems of dimension two over the ring of integers modulo pt

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