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Real and Hyperbolic Matrices of Split Semi Quaternions

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Yasemin Alagöz at Yıldız Technical University
Yasemin Alagöz
Gözde Özyurt at Beykent University
Gözde Özyurt
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Abstract

The aim of this paper is to investigate split semi quaternion matrices. To verify this, we first examine matrices with real split semi quaternion entries as a pair of hyperbolic matrices. Then, we present hyperbolic split semi quaternions and their matrices. Finally, for a hyperbolic split semi quaternion, we search some properties of its 2×22×22\times 2 matrix representation and we express 4×44×44\times 4 hyperbolic matrix and 8×88×88\times 8 real matrix representations.
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Adv. Appl. Clifford Algebras (2019) 29:53
c
2019 Springer Nature Switzerland AG
0188-7009/030001-20
published online May 29, 2019
https://doi.org/10.1007/s00006-019-0973-0
Advances in
Applied Clifford Algebras
Real and Hyperbolic Matrices of Split Semi
Quaternions
Yasemin Alag¨oz and G¨ozde ¨
Ozyurt∗
Communicated by Wolfgang Spr¨ossig
Abstract. The aim of this paper is to investigate split semi quaternion
matrices. To verify this, we first examine matrices with real split semi
quaternion entries as a pair of hyperbolic matrices. Then, we present
hyperbolic split semi quaternions and their matrices. Finally, for a hy-
perbolic split semi quaternion, we search some properties of its 2 ×2
matrix representation and we express 4×4 hyperbolic matrix and 8 ×8
real matrix representations.
Mathematics Subject Classification. Primary 15B33, Secondary 11R52.
Keywords. Split semi quaternion, Hyperbolic split semi quaternion, Split
semi quaternion matrix, Hyperbolic split semi quaternion matrix.
1. Introduction
Real quaternions or simply quaternions were described by Sir William Rowan
Hamilton in 1843. The set of quaternions is often denoted by H. Any element
of His generally written in the form q=q0e0+q1e1+q2e2+q3e3where
q0,q
1,q
2,q
3are real numbers and {e0,e
1,e
2,e
3}is a basis of both Hand R4.
As a set, quaternions Hare a four dimensional vector space over the real
numbers with addition, scalar multiplication and quaternion multiplication.
The basis element e0acts an identity and e1,e
2,e
3satisfy the following rules
e2
1=e2
2=e2
3=−1,
e1e2=−e2e1=e3,e
2e3=−e3e2=e1,e
3e1=−e1e3=e2.
It is obvious that His noncommutative. A well known fact about quater-
nions is any quaternion can be represented as 2 ×2 complex matrix through
the bijective transformation [1,2]. In [3], a quaternion matrix which en-
tries are quaternions have studied to a pair of complex matrices. Split semi
∗Corresponding author.
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
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