Some new linear representations of matrix quaternions with some applications

In this study, firstly, it was shown that the set of real quaternion matrices   MH n is a 4dimensional module over the real matrix ring   M n and 2-dimensional module over the complex matrix ring   M n. Moreover, some new properties of the real quaternion matrices were described. Then, matrix representations of the real quaternion matrices were found easily by Matlab. These matrices were also applied to find the inverse of the real quaternion matrices and inverse matrices were obtained easily with these matrices. In addition, some new properties for matrix representations of the real quaternion matrices were found. Also, the inverse of the 22  real quaternion block matrices was obtained by new methods. Finally, a new method to calculate the determinant of the 22  real quaternion matrices was found and the determinant of these matrices was calculated easily with Matlab application.

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“… REAL QUATERNION MATRICES We will find the inverse of 22  real quaternion block matrices. Theorem 6.1.…”

Section: The Inverse Of 22mentioning

confidence: 99%

“…Lin and Wang [21] completed a 22  block matrix of real quaternions with a partially specified inverse. Also in [22], the general partitioned linear representation form of matrix quaternions are obtained. Localization theorems are discussed for the left and right eigenvalues of block quaternion matrices in [23].…”

Section: Introductionmentioning

confidence: 99%

Some New Properties of The Real Quaternion Matrices and Matlab Applications

Nalbant

Yüce

2019

Cumhuriyet Science Journal

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“…The numerical solution of SLDE was published by [9,11]. Matrix notation of the quaternion vector space was analysed by [1,7]. The exact and clear description of numerical solution SQDE is presented in this contribution with the utilization of matrix formalism and C# programming language.…”

Section: Introductionmentioning

confidence: 99%

An algorithm for solving of Euler parameters differential equations system

Rédl¹

2021

MERAA

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The design an optimal numerical method for solving a system of ordinary differential equations simultaneously is described in this paper. System of differential equations was represented by a system of linear ordinary differential equations of Euler’s parameters called quaternions. The components of angular velocity were obtained by the experimental way. The angular velocity of the centre of gravity was determined from sensors of acceleration located in the plane of the centre of gravity of the machine. The used numerical method for solving was a fourth-order Runge-Kutta method. The stability of solving was based on the orthogonality of a direct cosine matrix. The numerical process was controlled on every step in numerical integration. The algorithm was designed in the C# programming language.

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“…The most critical feature of quaternions is that multiplying two quaternions is noncommutative. Quaternions have applications in graphics, flight dynamics, orbital mechanics of satellites, crystallographic texture analysis, and especially communication [2]. Chaotic systems based on quaternion variables have advantages over those based on real or complex variables in that the former provide secure communication because of (i) the increased number of dimensions needed to intercept communication and (ii) the feasibility of being implemented using capacitors and inductors.…”

Section: Introductionmentioning

confidence: 99%

A New Nine-Dimensional Chaotic Lorenz System with Quaternion Variables: Complicated Dynamics, Electronic Circuit Design, Anti-Anticipating Synchronization, and Chaotic Masking Communication Application

2019

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In this paper, a chaotic quaternion autonomous nonlinear structure is introduced and intends to be a contribution. It is the first nonlinear dynamical system with quaternion variables to be studied in the literature. With nine dimensions, the new system is a high-dimensional one. Several vital characteristics and features of this model are investigated, such as its Hamiltonian, symmetry, signal flow graph, dissipation, equilibriums and their stability, Lyapunov exponents, Lyapunov dimension, bifurcation diagrams, and chaotic behavior. A circuit implementation is designed to realize the new system, and a scheme is designed to achieve anti-anticipating synchronization (AAS) of two identical chaotic attractors with quaternion variables based on a Lyapunov function and active control. The concept of AAS is yet to be explored in the literature. A simulation experiment is designed and executed to illustrate the effectiveness of the acquired results. After synchronization, numerical outcomes are planned to explain the status variables and errors of these chaotic attractors to prove that AAS is achieved. The secure communication problem is studied based on the obtained events of the AAS of two identical nonlinear Lorenz systems with quaternion variables. AAS connecting the drive and response systems in chaotic systems with quaternion variables is the key to achieving communication. Signal encryption and restoration are simulated numerically.

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Some new linear representations of matrix quaternions with some applications

Cited by 15 publications

References 36 publications

Some New Properties of The Real Quaternion Matrices and Matlab Applications

Some New Properties of The Real Quaternion Matrices and Matlab Applications

An algorithm for solving of Euler parameters differential equations system

A New Nine-Dimensional Chaotic Lorenz System with Quaternion Variables: Complicated Dynamics, Electronic Circuit Design, Anti-Anticipating Synchronization, and Chaotic Masking Communication Application

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