Some new properties of octonionic matrices

Abstract

Keywords: Octonion; Matrix; Module

Introduction

Octonions are non-associative algebras. They form the largest normed division algebra. The octonions were discovered independently by Graves and Cayley [1]. The set of octonions can be written in the form:

where ai’s are real numbers (coefficients of octonions), ei’s (0 ≤ j ≤ 7)are the octonion units (basis elements of octonions), and e0 = +1 is the multiplicative scalar element. These octonion units satisfy the following properties:

where is the usual Kronecker delta symbol and are completely antisymmetric tensor and they are equal to 1 for following combinations [2]:

Dray and Manogue discussed the eigenvalue problem for 2 × 2 and 3 × 3 octonionic Hermitian matrices. In both cases, they gave the general solution for real eigenvalues, and they showed there are also solutions with non-real eigenvalues [3]. Dray and Manogue described the use of Mathematica in analyzing octonionic eigenvector problem,

*Corresponding author. *E-mail address: ozcan.bektas@samsun.edu.tr, sayuce@yildiz.edu.tr This paper was recommended for publication in revised form by Regional Editor Mostafa Safdari Shadloo Published by Yıldız Technical University Press, İstanbul, Turkey Copyright 2021, Yıldız Technical University. This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

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and in particular its use in proving a generalized orthogonality property for which no other proof is known [4]. The eigenvalue problem of symmetric 3×3 octonionic matrix has been analyzed by Okubo [5]. Gillow-Wiles and Dray showed that any 3-component octonionic vector which is purely imaginary, but not quaternionic, is an eigenvector of a 6-parameter family of Hermitian octonionic matrices, with imaginary eigenvalue equal to the associator of its elements [9]. Serôdio et.al. studied 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 [12]. Octonions and octonionic matrices have applications in fields such as string theory, special relativity and quantum logic. Tian gave a complete investigation to real matrix representations of octonions, and considered their various applications to octonions as well as matrices of octonions [6]. Daboul and Delbourgo defined a special matrix multiplication among a special subset of 2N×2N matrices, and studied the resulting (non-associative) algebras and their subalgebras. They derived the conditions under which these algebras become alternative non-associative and when they become associative [13]. The determinant of octonionic matrices and its properties were given by Li and Yuan [7]. Nieminen gave twoby-two random matrix theory with matrix representations of octonions [8]. Karataş and Halıcı investigated octonions and their special vector matrix representation. Theye gave some geometrical definitions and properties related with them [10]. Split-type octonion matrix was given by Bektaş, [11].

The set of octonions is denoted by 𝕆 . The sum operation on this set is defined as follows:

Octonions Let us first give some fundamental notions of the octonions. The real octonion A is defined by , where ai’s are the real number components of the octonions, are the unit octonions basis elements,and e0 = +1 is the scalar element [14]. The multiplication rules of these unit octonion basis elements are given by :

Another operation on the set of octonion is the conjugate operation. The conjugate of the octonion A is denoted by A and is defined as follows:

where and , [15]. Besides that, the octonion A has real part and vectorial part. They are called the real (SA), and vectorial (VA) parts of the octonion A, [15, 16]:

Thus, an octonion A can be written by A = SA + VA . The multiplication of A, B ∈ 0 ,is defined by (1)

is symmetric, non-degenerate real-valued bilinear form and is called the octonionic inner product. If , then the octonion A is called the spatial (pure) octonion. The norm of the octonion A is denoted by

If , then A0 is called the unit octonion [15][17]. The inverse of an octonion is defined by

Table 1.The multiplication table of the unit cctonion basis elements If A and B octonions, then , [20].

Some New Properties of Octonionic Matrices In this Section, we will investigate some new properties of octonion matrices. We can list some of these new properties as follows: 1) Real, complex and quaternion coefficient matrices representations of octonionic matrices. 2) Basic operations on octonionic matrices.

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3) Conjugate, transpose, conjugate transpose, inverse and trace of octonionic matrices. 4) Algebraic structures of the set of octonionic matrices. 5) Special defined octonionic matrix and their properties.

Definition 6 Let be an octonionic matrix. The octonionic matrix as a combination of two quaternionic matrices is written as

where Definition 7 Let given. If , then it is called written . Remark 1 Let

Let be two octonionic matrices. The addition operation of octonionic matrices as follows:

Definition 3 Let and be an octonionic matrix, then the multiplication of a real number and an octonionic matrix defined as follows and

Definition 4 Let be an octonionic matrix. The octonionic matrix is written in real combination as

The Properties of the Addition Operation of the Octonionic Matrices Let and , then the following properties are satisfied

be an octonionic matrix. The octonionic matrix as a combination of four complex matrices is written as

be two octonionic matrices. The multiplication of the octonionic matrices defined by where

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On the other hand, we can write the multiplication as follows:

Here, and real matrices. Then, the multiplication operation in terms of the real combination of two octonionic matrices can be defined as follows: and

ionic matrices. Let’s find the product of these matrices and calculate it in terms of real, complex and quaternion combinations. According to the definition of multiplication , we get

The multiplication operation in terms of the real combination of two octonionic matrices and can be defined as follows:

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and Theroem 2 (The Properties of Conjugate) The multiplication operation in terms of the complex combination of two octonionic matrices A and B can be defined as follows:

Proof. (1), (2), and (3) can be easily shown. Now we will prove one condition of (4): 4) Similarly, the product of and can be calculated in terms of quaternion combinations. Theorem 1(The Properties of the Product of the Octonionic Matrices ) 1) Let be given.

Conjugate of Octonionic Matrices Definition 10 Let be an octonionic matrix. Then, the conjugate of octonionic matrix is defined as . Remark 5 The conjugate of octonionic matrix . Remark 6 The conjugate of octonionic matrix is . Proof. Let , then we get

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Remark 8 Let given. Then, the transpose of octonionic matrix . Proof. For , then we get

be given. Then, the conjugate transpose of octon. be conjugate

Thereom 4 (The Properties of Conjugate Transpose Operation) be Let given. Then, the following properties are hold: 10

be given. Then, the transpose of octonionic matrix . Theroem 3 (The Properties of Transpose Operation) and

Proof. (2), (3), and (4) can be easily shown. Now we will prove one condition of (1): 1) From the 2-th property of the conjugate operation, So, we get

Proof. (1), (2), and (4) can be easily shown. Now we will prove one condition of (3): 3) Let be given. Then we get

given. The sum of the elements of the octonionic square matrix A on the principal diagonal is called the trace of the matrix

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. If we take conjugate transpose of both side of the equation Proof. (1), (3), and (4) can be easily shown. Now we will prove one condition of (2): 2) We know that

Similarly, if we take conjugate transpose of both side of the equation , we get

The properties of the trace function of real matrix is . So, the equations (2) and (3) is not equal. Finally, we get

From the definition of the Hermitien matrix, we get and Remark

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Algebraic Structures of the Set of Octonionic Matrices Vector space structure on Definition 17 Let

be a real octonionic matrix. Then the multiplication of real number and a real octonionic matrix is defined as

Definition 19 Let Let and be unit element for .Then the multiplication of real number and a real octonionic matrix provides the following properties:

multiplication of a quaternion and a octonion matrix is defined as

lar multiplication operations are scaler operations. These operations are not equal. So, in general

be octonionic matrices, and Corollarly 4 The basis of the vector space the following set S1.

. Then, the left product of a square real matrix and an square octonionic matrix is defined as Corollarly 5 Let the set be given. 1) The system S1 is linear independent, 2) Because of the above corollarly, the matrix system S1 is standart basis of the vector space . So, Left (right) module structure on Definition 18 Let ionic matrix and

multiplication of a complex number and a octonion matrix is defined as or

lar multiplication operations are scaler operations. These operations are not equal. So, in general

Let and be unit element of . Then the multiplication of real matrix and a real octonionic matrix provides the following properties:

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Corollarly 8 space. The basis of the vector space is the following set S2.

defined for each definition of octonionic matrices. Finally, algebraic structures of set of octonionic matrices have been searched. The objective of this study is to research octonionic matrix and their properties. The octonic matrix concept can also be studied using sedenions.

Authorship Contributions

Corollarly 9 Let the set be given. 1) The system S2 is linear independent, 2) Because of the above corollarly, the matrix system S2 is standart basis of the vector space . So, Left (right) module structure on Definition

. Then, the left product of a square complex matrix and an square octonionic matrix is defined as

Data Availability Statement

The authors confirm that the data that supports the findings of this study are available within the article. Raw data that support the finding of this study are available from the corresponding author, upon reasonable request.

Conflict Of Interest

The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Ethics

Since the octonionic multiplication operation does not provide the associative property, then is not a left(right) module. Similarly, the right product of a square complex matrix and a square octonionic matrix can be defined. Left (right) module structure on Definition

There are no ethical issues with the publication of this manuscript.

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