1. Introduction
An octonion number
can be expressed as:
(1)
where
are real numbers and
are imaginary units. Their multiplication is given in Table 1. Octonion algebra is an eight-dimensional, non-commutative, non-associative and normed division algebra.
Table 1. Imaginary units multiplication.
Octonions have been used in many fields of mathematics, and they have many applications, about octonions and their applications see [1] [2] [3], and [4] [5] to take a historical overview.
In the matrix representation, an octonion
can be represented by
real matrices. One form of these matrices is a matrix A (the left matrix representation) [6]:
The problem of computing the nth power of an octonion is still interesting to many researchers. Some methods are used, such as binomial expansion, De Moivre’s formula, and Euler’s formula. To solve this problem, we use a new technique to construct formulas computing the powers of an octonion.
2. The Methodology
For a complex number
where
are real numbers, and i is the imaginary unit satisfies
,
, n is a positive integer number (2)
If n is an even number then there will be
real terms and
imaginary
terms, to simplify, we define
,
and, let
be nonzero real numbers, we can write (2) as:
(3)
If n is an odd number then there will be
real terms and
imaginary
terms, we can write (2) as:
(4)
To prove (3), let us write b in a matrix representation form:
For
,
can be computed from:
,
so
(5)
Assume that (3) is true for
, where k is a positive integer, therefore:
(6)
We can compute
by using the matrix:
Multiplying the first row of
by
(the column matrix representing
) gives
, and multiplying the second row of
by
gives
.
By the similar way, we can prove (4).
3. Results
We can use (3) and (4) to compute the powers of a quaternion number and the powers of an octonion number.
To compute the power of
, replace
by
and
by
in (3) and (4),
(7)
(8)
(7) and (8) give
when n is an even number and n is an odd number respectively.
Of course the proof is similar to that one we used to prove the powers formulas for a complex number, but to make it clearer, we will write A as:
(9)
where
,
Since
will play an important role in our proof, we call it the fundamental matrix. The following proposition presents the main properties of the fundamental matrix
.
3.1. Proposition
Let
,
be the conjugate of
,
be a real number, and
be the identity matrix. Then
(a)
(b)
(c)
(d)
(e)
Proof:
(a)
, (since
).
In general
,
, (n is a real number).
(c)
The verification of remaining propositions is straightforward.
Form proposition (a),
can be computed from:
So, multiplying the first row of
by
gives
, and to obtain
we just multiply any row (except the first row) of
by
because the multiplication result will be similar for all other rows, therefore
will be summation of these results.
In general, since
, we will obtain
by multiply the first row of
by
, and to obtain
we multiply any row (except the first row) of
by
.
In case
(
is a pure octonion) and
, if n is an even number then:
(10)
And if n is an odd number then:
(11)
3.2. Example
Let
From (10),
So,
From (11),
3.3. Example
To compare our formulas with the De Moivre’s formula and Euler’s formula that were used in [7] to find
for
, take
From (8),
So,
, and
Also let us compute
.
From (7),
So,
, and
4. Conclusion
The formulas presented in this work are more suitable for computing the powers of an octonion number (the powers of matrices representing an octonion number). These formulas which are derived from binomial expansion also can be used to compute the power of a quaternion number (the powers of matrices representing a quaternion number), and a complex number (the powers of matrices representing a complex number).