Matrix representations of octonions and their applications

@article{Tian2000MatrixRO,
  title={Matrix representations of octonions and their applications},
  author={Yongge Tian},
  journal={Advances in Applied Clifford Algebras},
  year={2000},
  volume={10},
  pages={61-90}
}
  • Yongge Tian
  • Published 2000
  • Mathematics
  • Advances in Applied Clifford Algebras
As is well-known, the real quaternion division algebra ℍ is algebraically isomorphic to a 4-by-4 real matrix algebra. But the real division octonion algebra can not be algebraically isomorphic to any matrix algebras over the real number field ℝ, because is a non-associative algebra over ℝ. However since is an extension of ℍ by the Cayley-Dickson process and is also finite-dimensional, some pseudo real matrix representations of octonions can still be introduced through real matrix… Expand
Hermitian randommatrices with octonion entries ( a ) Preliminaries
  • 2017
Octonions in random matrix theory
  • P. Forrester
  • Mathematics, Physics
  • Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
  • 2017
The octonions are one of the four normed division algebras, together with the real, complex and quaternion number systems. The latter three hold a primary place in random matrix theory, where inExpand
The Minimal Polynomial of Some Matrices Via Quaternions
This work provides explicit characterizations and formulae for the minimal polynomials of a wide variety of structured 4 × 4 matrices. These include symmetric, Hamiltonian and orthogonal matrices.Expand
An Algebraic Model for Real Matrix Representations. Remarks Regarding Quaternions and Octonions
In this chapter, we present some applications of quaternions and octonions. We present the real matrix representations for complex octonions and some of their properties which can be used inExpand
Some equations over generalized quaternion and octonion division algebras
It is known that any polynomial of degree n with coefficients in a field K has at most n roots in K. If the coefficients are inH (the quaternion algebra), the situation is different. For H over theExpand
Minimal Polynomials of Some Matrices Via Quaternions
This work provides explicit characterizations and formulae for the minimal polynomials of a wide variety of structured $4\times 4$ matrices. These include symmetric, Hamiltonian and orthogonalExpand
Eigenvalues of matrices related to the octonions
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 theExpand
Construction of Octonionic Polynomials
Abstract.In a previous paper “[On Octonionic Polynomials”, Advances in Applied Clifford Algebras, 17 (2), (2007), 245–258] we discussed generalizations of results on quaternionic polynomials to theExpand
EIGENVALUES AND EIGENVECTORS FOR THE QUATERNION MATRICES OF DEGREE TWO
In this paper we give a computation method, in a particular case, for eigenvalues and eigenvectors of the quaternion matrices of degree two with elements in the generalized quaternion divisionExpand
Quaternions, octonions, and now, 16-ons and 2 n -ons; New kinds of numbers
“Cayley-Dickson doubling,” starting from the real numbers, successively yields the complex numbers (dimension 2), quaternions (4), and octonions (8). Each contains all the previous ones asExpand
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 27 REFERENCES
Matrix representation of octonions and generalizations
We define a special matrix multiplication among a special subset of 2N×2N matrices, and study the resulting (nonassociative) algebras and their subalgebras. We derive the conditions under which theseExpand
Eigenvalue problem for symmetric 3×3 octonionic matrix
The eigenvalue problem of symmetric 3×3 octonionic matrix has been analyzed. We have especially proved explicitly first that octonionic eigenfunctions have six independent solutions in general withExpand
Similarity and consimilarity of elements in the real Cayley-Dickson algebras
AbstractThe similarity and consimilarity of elements in the real quaternion, octonion and sedenion algebras, as well as in the general real Cayley-Dickson algebras are considered by solving the twoExpand
Introduction to octonion and other non-associative algebras in physics
1. Introduction 2. Non-associative algebras 3. Hurwitz theorems and octonions 4. Para-Hurwitz and pseudo-octonion algebras 5. Real division algebras and Clifford algebra 6. Clebsch-Gordon algebras 7.Expand
Universal similarity factorization equalities over real Clifford algebras
A variety of universal similarity factorization equalities over real Clifford algebrasRp,q are established. On the basis of these equalities, real, complex and quaternion matrix representations ofExpand
Division Algebras:: Octonions Quaternions Complex Numbers and the Algebraic Design of Physics
I. Underpinnings. II. Division Algebra Alone. III. Tensor Algebras. IV. Connecting to Physics. V. Spontaneous Symmetry Breaking. VI. 10 Dimensions. VII. Doorways. VIII. Corridors. Appendices.Expand
Universal Similarity Factorization Equalities Over Complex Clifford Algebras
A set of valuable universal similarity factorization equalities is established over complex Clifford algebras C n. Through them matrix representations of complex Clifford algebras C n can directly beExpand
The octonionic eigenvalue problem
We discuss the eigenvalue problem for 2×2 and 3×3 octonionic Hermitian matrices. In both cases, we give the general solution for real eigenvalues, and we show there are also solutions with non-realExpand
Manogue, The octonionic eigenvalue problem, Adv
  • Appl. Clifford Algebras,
  • 1998
Universal factorization equalities over real Clifford algebras
  • Adv. Appl. Clifford Algebras
  • 1998
...
1
2
3
...