Real matrix representations of complex split quaternions with applications

2020 ◽  
Vol 43 (12) ◽  
pp. 7227-7238
Author(s):  
Melek Erdoğdu ◽  
Mustafa Özdemir
Keyword(s):  
Real Matrix ◽  
Matrix Representations ◽  
Split Quaternions

scholarly journals Expression of a Real Matrix as a Difference of a Matrix and its Transpose Inverse

2019 ◽  
Vol 11 (1) ◽  
pp. 1-6
Author(s):  
Mil Mascaras ◽  
Jeffrey Uhlmann
Keyword(s):  
Control Theory ◽  
Computational Simulation ◽  
Real Matrix ◽  
Matrix Representations ◽  
Articulated Figures ◽  
The Difference

In this paper we derive a representation of an arbitrary real matrix M as the difference of a real matrix A and the transpose of its inverse. This expression may prove useful for progressing beyond known results for which the appearance of transpose-inverse terms prove to be obstacles, particularly in control theory and related applications such as computational simulation and analysis of matrix representations of articulated figures.

scholarly journals Eigenvalues of matrices related to the octonions

4open ◽  
2019 ◽  
Vol 2 ◽  
pp. 16
Author(s):  
Rogério Serôdio ◽  
Patricia Beites ◽  
José Vitória
Keyword(s):  
Matrix Representation ◽  
Real Matrix ◽  
Matrix Representations ◽  
Estimation Algorithms ◽  
The Matrix

A pseudo real matrix representation of an octonion, which is based on two real matrix representations of a quaternion, is considered. We study how some operations defined on the octonions change the set of eigenvalues of the matrix obtained if these operations are performed after or before the matrix representation. The established results could be of particular interest to researchers working on estimation algorithms involving such operations.

scholarly journals Real Matrix Representations for the Complex Quaternions

2013 ◽  
Vol 23 (3) ◽  
pp. 657-671 ◽  
Author(s):  
Cristina Flaut ◽  
Vitalii Shpakivskyi
Keyword(s):  
Real Matrix ◽  
Matrix Representations

scholarly journals On the Solutions of Some Linear Complex Quaternionic Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Cennet Bolat ◽  
Ahmet İpek
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Sylvester Equation ◽  
Real Matrix ◽  
Matrix Representations ◽  
Linear Complex ◽  
The Real ◽  
Special Cases ◽  
Necessary And Sufficient

Some complex quaternionic equations in the typeAX-XB=Care investigated. For convenience, these equations were called generalized Sylvester-quaternion equations, which include the Sylvester equation as special cases. By the real matrix representations of complex quaternions, the necessary and sufficient conditions for the solvability and the general expressions of the solutions are obtained.

scholarly journals ON THE REAL MATRIX REPRESENTATION OF PT-SYMMETRIC OPERATORS

2014 ◽  
Vol 54 (2) ◽  
pp. 113-115 ◽  
Author(s):  
Francisco M. Fernández
Keyword(s):  
Matrix Representation ◽  
Real Matrix ◽  
Matrix Representations ◽  
The Real ◽  
Symmetric Operators

We discuss the construction of real matrix representations of PT-symmetric operators. We show the limitation of a general recipe presented some time ago for non-Hermitian Hamiltonians with antiunitary symmetry and propose a way to overcome it. Our results agree with earlier ones for a particular case.

scholarly journals A Note on the Representation of Clifford Algebra

2020 ◽  
Author(s):  
Ying-Qiu Gu
Keyword(s):  
Clifford Algebra ◽  
Practical Calculation ◽  
Real Matrix ◽  
Matrix Representations ◽  
Normal Representation ◽  
Riemann Geometry ◽  
Minkowski Space Time ◽  
Pauli Matrices ◽  
Dimensional Minkowski Space ◽  
Real Clifford Algebra

In this note we construct explicit complex and real matrix representations for the generators of real Clifford algebra $C\ell_{p,q}$. The representation is based on Pauli matrices and has an elegant structure similar to the fractal geometry. We find two classes of representation, the normal representation and exceptional one. The normal representation is a large class of representation which can only be expanded into $4m+1$ dimension, but the exceptional representation can be expanded as generators of the next period. In the cases $p+q=4m$, the representation is unique in equivalent sense. These results are helpful for both theoretical analysis and practical calculation. The generators of Clifford algebra are the faithful basis of $p+q$ dimensional Minkowski space-time or Riemann space, and Clifford algebra converts the complicated relations in geometry into simple and concise algebraic operations, so the Riemann geometry expressed in Clifford algebra will be much simple and clear.

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