| Issue |
4open
Volume 2, 2019
Advances in Researches of Quaternion Algebras
|
|
|---|---|---|
| Article Number | 16 | |
| Number of page(s) | 7 | |
| Section | Mathematics - Applied Mathematics | |
| DOI | https://doi.org/10.1051/fopen/2019014 | |
| Published online | 05 June 2019 | |
Research Article
Eigenvalues of matrices related to the octonions
1
Departamento de Matemática and CMA-UBI, Universidade da Beira Interior, 6201-001 Covilhã, Portugal
2
Departamento de Matemática, Universidade de Coimbra, 3004-531 Coimbra, Portugal
* Corresponding author: rserodio@ubi.pt
Received:
29
December
2018
Accepted:
20
April
2019
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.
Mathematics Subject Classification: 11R52 / 15A18
Key words: Octonions / Quaternions / Real matrix representations / Eigenvalues
© R. Serôdio et al., Published by EDP Sciences, 2019
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
Due to nonassociativity, the real octonion division algebra is not algebraically isomorphic to a real matrix algebra. Despite this fact, pseudo real matrix representations of an octonion may be introduced, as in [1], through real matrix representations of a quaternion.
In this work, the left matrix representation of an octonion over 
, as called by Tian in [1], is considered. For the sake of completeness, some definitions and results, in particular on this pseudo representation, are recalled in Section 2.
Using the mentioned representation, results concerning eigenvalues of matrices related to the octonions are established in Section 3. Previous research on this subject, although not explicitly applying real matrix representations of a quaternion, can be seen in [2].
2 Real octonion division algebra
Consider the real octonion division algebra 
, that is, the usual real vector space 
, with canonical basis 
, equipped with the multiplication given by the relations
where δ
ij
is the Kronecker delta, ε
ijk
is a Levi-Civita symbol, i.e., a completely antisymmetric tensor with a positive value +1 when ijk = 123, 145, 167, 246, 275, 374, 365 and
e
0
is the identity. This element will be omitted whenever it is clear from the context.
Every element
o
∈ 
can be written as 
where Re(
o
) = o
0 and 
are called the real part and the imaginary (or vector) part, respectively. The conjugate of
o
is defined as 
. The norm of
o
is defined by 
. The inverse of a non-zero octonion
o
is 
.
The multiplication of 
can be written in terms of the Euclidean inner product and the vector cross product in 
, hereinafter denoted by • and ×, respectively. Concretely, as in [3], we have 
Following [4], we recall that 
are perpendicular if 
. In particular, if 
, then 
are perpendicular if 
. Moreover, 
are parallel if 
. In particular, if 
, then 
are parallel if 
.
The elements of the basis of 
can also be written as 
The real octonion division algebra 
, of dimension 8, can be constructed from the real quaternion division algebra 
, of dimension 4, by the Cayley-Dickson doubling process where 
contains 
as a subalgebra. As a consequence, it is well known that any 
can be written as 
(1) where 
are of the form 
, with 
.
The real quaternion division algebra 
is algebraically isomorphic to the real matrix algebra of the matrices in (2), where ϕ(q) is a real matrix representation of a quaternion q.
Definition 1
Some important properties of the matrices in Definition 1 are recalled in Lemma 1.
Lemma 1
[1] Let 
and 
. Then
-
. -
. -
. -
, if 
. -
.
The real quaternion division algebra 
is algebraically anti-isomorphic to the real matrix algebra of the matrices in (3), where τ(q) is another real matrix representation of a quaternion q.
Definition 2
[1] Let 
. Then
(3) where K4 = diag(1, −1, −1, −1).
Some relevant properties of the matrices in Definition 2 are recalled in Lemma 2.
Lemma 2
[1] Let 
and 
. Then
-
. -
τ(a + b) = τ(a) + τ(b), τ(ab) = τ(b)τ(a), τ(λa) = λτ(a), τ(1) = I 4.
-
. -
, if 
. -
.
Due to the non-associativity, the octonion algebra cannot be isomorphic to the real matrix algebra with the usual multiplication. With the purpose of introducing a convenient matrix multiplication, we show another way of representing the octonions by a column matrix.
Definition 3
Let 
. The column
, vectorial or ket representation of
o
is 
.
Based on the previous real matrix representations of a quaternion, Tian introduced the following pseudo real matrix representation of an octonion.
Definition 4
[1] Let 
, where 
. Then the 8 × 8 real matrix
(4) is called the left matrix representation of
a
over 
, where K
4 = diag(1, −1, −1, −1).
The meaning of the term left matrix representation comes from the following result.
Theorem 1
[1] Let 
. Then 
.
Tian [1] introduced also the right matrix representation of an octonion
a
over 
, which he denoted by 
. In this case, Tian proved that 
.
Even though there are 
such that 
, there are still some properties which hold. These are recalled in Theorem 2.
Theorem 2
[1] Let 
, 
. Then
-
. -
ω( a + b ) = ω( a ) + ω( b ), ω(λ a ) = λω( a ), ω(1) = I 8.
-
.
3 Main results
In this section, the left matrix representation of an octonion over 
is considered. First of all, given an octonion, the eigenvalues of its left matrix representation are computed.
Proposition 1
Let 
. Then the eigenvalues of the real matrix ω(
a
) are
each with algebraic multiplicity 4.
Proof. Given an octonion a, it can always be uniquely represented as 
such that 
, and where 
. The characteristic polynomial of ω(
a
) is 
where K
8 is the orthogonal matrix diag(1, −1, −1, −1, 1, 1, 1, 1) = diag(K
4, I
4). Hence, 
and the result follows.
The set of eigenvalues of ω( ab ) is equal to the set of eigenvalues of ω( a )ω( b ) since the characteristic polynomials are equal as can easily be seen. However, if we add an extra octonion c the set of eigenvalues of ω( ab + c ) and ω( a )ω( b ) + ω( c ) may differ.
We now study the eigenvalues of the matrix ω( a )ω( b ) + ω( c ), given three octonions a , b , and c .
Remark 1
Let 
and 
. Notice that 
can be decomposed into two parts: a part parallel to 
, denoted by 
; and a part perpendicular to 
, denoted by 
. The parallel part is the projection of 
onto 
, which is defined as
The perpendicular part is given by 
.
Besides the projection of 
onto 
we will also consider the projection of 
over 
, i.e., the linear space generated by all linear combinations of 
and 
.
Remark 2
As 
and 
, then
where 
and 
. Hence, it suffices to consider only the product 
since the remaining terms can be added to
c
.
The pure octonion 
can be decomposed in two parts, one in 
and the other perpendicular to it. In this case, we can write 
, where 
and 
. The parallel part is the projection of 
onto 
and is given by
and the perpendicular part is naturally 
.
Proposition 2
Let 
such that 
, and the imaginary part of
a
and
b
are perpendicular. Then the eigenvalues of the real matrix ω(
a
)ω(
b
) + ω(
c
) are
(5)
where
c
∥ is the projection of
c
onto Span(
a
,
b
) and 
, each with algebraic multiplicity 2.
Proof. Without loss of generality, we consider
a
= a
i
and
b = b
j
. Hence, 
By Lemmas 1 and 2, we have 
(6) Let 
, where 
. Then 
(7)
Taking into account (6) and (7), we obtain 
The characteristic polynomial of ω(
a
)ω(
b
) + ω(
c
) is 
where K8 is the orthogonal matrix diag(1, −1, −1, −1, 1, 1, 1, 1) = diag(K
4, I
4). Hence, 
which results in 
and, since 
, gives 
where 
and 
. By Lemma 2, we have 
and the result follows.
The following corollary may be useful to improve the localization of eigenvalues of octonionic matrices and zeros of octonionic polynomials whenever such products occur.
Corollary 2.1
Let 
. Then
(8) where ρ(•) stands for the spectral radius.
Proof. By Proposition 2, we obtain 
Furthermore, the eigenvalues of ω(
ab +
c
) are all equal in modulus and satisfy 
Without loss of generality, we can consider
a
= a
i
and
b
= b
j
. If 
. Hence, we arrive at 
and the result follows.
Example 3.1
To apply (5)
, we have to take into account Remarks 1 and 2, and rewrite
a
,
b
and
c
as
where 
, 
and 
are the imaginary parts of
a
,
b
and
c
, respectively.
Computing 
, the projection of 
onto 
, we obtain 
. Thus, the orthogonal part 
is equal to 
.
The projections of 
onto 
and 
are, respectively 
and 
. Hence, the projection of 
on the space of 
and 
is 
. This implies that for the orthogonal part we have 
.
Taking all together, we have
where 
is real and 
. So
(9) where 
, 
, and 
.
We are now in condition to apply Proposition 2, from which we obtain the eigenvalues of
The eigenvalues are 
and 
, while the eigenvalues of 
are 
.
As predicted by Corollary 2.1, 
, since 
and 
.
Acknowledgments
The authors are very thankful to the anonymous referee for his valuable suggestions towards the improvement of this paper.
R. Serôdio and P.D. Beites were supported by Fundação para a Ciência e a Tecnologia (Portugal), project UID/MAT/00212/2013 of the Centro de Matemática e Aplicações da Universidade da Beira Interior (CMA-UBI). P.D. Beites was also supported by the research project MTM2017-83506-C2-2-P (Spain).
References
- Tian Y (2000), Matrix representations of octonions and their applications. Adv Appl Clifford Algebras 10, 61–90. [CrossRef] [Google Scholar]
- Beites PD, Nicolás AP, Vitória J (2017), On skew-symmetric matrices related to the vector cross product in R7. Electron J Linear Algebra 32, 138–150. [CrossRef] [Google Scholar]
- Leite FS (1993), The geometry of hypercomplex matrices. Linear Multilinear Algebra 34, 123–132. [CrossRef] [Google Scholar]
- Ward JP (1997), Quaternions and Cayley numbers, Kluwer, Dordrecht. [CrossRef] [Google Scholar]
Cite this article as: Serôdio R, Beites P & Vitória J 2019. Eigenvalues of matrices related to the octonions. 4open, 2, 16.
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