|
In mathematics, a biquaternion is a numeric and geometric concept developed by
William Kingdon Clifford, William Rowan Hamilton, and Alexander MacAuley in the nineteenth century.
Let {1, i, j, k} be the basis for the (real) quaternions, and let u, v, w, x be Complex numbers, then q = u 1 + v i + w j + x k is a biquaternion.
The collection of all biquaternions forms a vector space of four complex dimensions or eight real dimensions.
Considered with the operations of matrix addition and multiplication, this collection forms a non-commutative but associative ring.
Linear Representation
Note the matrix product
- <math>\begin{pmatrix}i & 0\\0 & -i\end{pmatrix}\begin{pmatrix}0 & 1\\-1 & 0\end{pmatrix}<math> = <math>\begin{pmatrix}0 & i\\i & 0\end{pmatrix}<math>
where each of these three arrays has a square equal to the negative of the identity matrix.
When the matrix product is interpreted as i j = k, then one obtains a subgroup of the group matricies that is isomorphic to the Quaternion group.Consequently
- <math>\begin{pmatrix}u+iv & w+ix\\-w+ix & u-iv\end{pmatrix}<math> represents biquaternion q.
Given any 2x2 complex matrix, there are complex values u, v, w, and x to put it in this form so that the matrix ring is isomorphic to the biquaternion ring.
Alternative Complex Plane
Suppose we take w to be purely imaginary, w = b ι , where ι ι = - 1 .(Here one uses iota instead of i for the complex imaginary to be distinct from quaternion i.)
Now when r = w j , then its square is
r r = (w j )(w j ) = (w w)(j j ) = b b (-1)(-1) = b2 .
In particular, when b = 1 or –1, then r 2 = + 1 . This development shows that
the biquaternions are a source of "algebraic motors" like r that square to +1. Then {a + b
ι j : a, b ∈ R } is a subring of biquaternions isomorphic to the split-complex number ring.
|
