# 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} }

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

#### 69 Citations

Octonions in random matrix theory

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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 in… Expand

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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 in… Expand

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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 the… Expand

Minimal Polynomials of Some Matrices Via Quaternions

- Mathematics, Physics
- 2010

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 orthogonal… Expand

Eigenvalues of matrices related to the octonions

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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… Expand

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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 division… Expand

Quaternions, octonions, and now, 16-ons and 2 n -ons; New kinds of numbers

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“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 as… Expand

#### References

SHOWING 1-10 OF 27 REFERENCES

Matrix representation of octonions and generalizations

- Mathematics, Physics
- 1999

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 these… Expand

Eigenvalue problem for symmetric 3×3 octonionic matrix

- Mathematics
- 1999

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 with… Expand

Similarity and consimilarity of elements in the real Cayley-Dickson algebras

- Mathematics, Physics
- 1999

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 two… Expand

Introduction to octonion and other non-associative algebras in physics

- Physics, Mathematics
- 1995

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

- Mathematics, Physics
- 1998

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 of… Expand

Division Algebras:: Octonions Quaternions Complex Numbers and the Algebraic Design of Physics

- Mathematics
- 1994

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

- Mathematics, Physics
- 2000

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 be… Expand

The octonionic eigenvalue problem

- Mathematics
- 1998

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-real… Expand

Manogue, The octonionic eigenvalue problem, Adv

- Appl. Clifford Algebras,
- 1998

Universal factorization equalities over real Clifford algebras

- Adv. Appl. Clifford Algebras
- 1998