Quaternion

A member of a noncommutative Division Algebra first invented by William Rowan Hamilton. The quaternions are sometimes also known as Hypercomplex Numbers and denoted $\Bbb{H}$. While the quaternions are not commutative, they are associative.

The quaternions can be represented using complex $2\times 2$ Matrices

$$H=\left[{\matrix{z & w\cr -w^* & z^*\cr}}\right] = \left[{\matrix{a+ib & c+id\cr -c+id & a-ib\cr}}\right],$$ (1)

where $z$ and $w$ are Complex Numbers, $a$, $b$, $c$, and $d$ are Real, and $z^*$ is the Complex Conjugate of $z$. By analogy with the Complex Numbers being representable as a sum of Real and Imaginary Parts, $a\cdot 1+bi$, a quaternion can also be written as a linear combination

$$H = a{\hbox{\sf U}} + b{\hbox{\sf I}} + c{\hbox{\sf J}} + d{\hbox{\sf K}}$$ (2)

of the four matrices

$${\hbox{\sf U}} \equiv \left[\begin{array}{cc}1 & 0\\ 0 & 1\end{array}\right]$$ (3)

$${\hbox{\sf I}} \equiv \left[\begin{array}{cc}i & 0\\ 0 & -i\end{array}\right]$$ (4)

$${\hbox{\sf J}} \equiv \left[\begin{array}{cc}0 & 1\\ -1 & 0\end{array}\right]$$ (5)

$${\hbox{\sf K}} \equiv \left[\begin{array}{cc}0 & i\\ i & 0\end{array}\right].$$ (6)

(Note that here, ${\hbox{\sf U}}$ is used to denote the Identity Matrix, not ${\hbox{\sf I}}$.) The matrices are closely related to the Pauli Spin Matrices $\sigma_x$, $\sigma_y$, $\sigma_z$, combined with the Identity Matrix. From the above definitions, it follows that

$${\hbox{\sf I}}^2 = -{\hbox{\sf U}}$$ (7)

$${\hbox{\sf J}}^2 = -{\hbox{\sf U}}$$ (8)

$${\hbox{\sf K}}^2 = -{\hbox{\sf U}}.$$ (9)

Therefore ${\hbox{\sf I}}$, ${\hbox{\sf J}}$, and ${\hbox{\sf K}}$ are three essentially different solutions of the matrix equation

$${\hbox{\sf X}}^2 = -{\hbox{\sf U}},$$ (10)

which could be considered the square roots of the negative identity matrix.

In $\Bbb{R}^4$, the basis of the quaternions can be given by

$$i \equiv \left[\begin{array}{cccc}0 & 1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & -1 & 0\end{array}\right]$$ (11)

$$j \equiv \left[\begin{array}{cccc}0 & 0 & 0 & -1 \\ 0 & 0 & -1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0\end{array}\right]$$ (12)

$$k \equiv \left[\begin{array}{cccc}0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0\end{array}\right]$$ (13)

$$1 \equiv \left[\begin{array}{cccc}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1\end{array}\right].$$ (14)

The quaternions satisfy the following identities, sometimes known as Hamilton's Rules,

$$i^2 = j^2 = k^2 = -1$$ (15)

$$ij = -ji = k$$ (16)

$$jk = -kj = i$$ (17)

$$ki = -ik = j.$$ (18)

They have the following multiplication table.

	1	$i$	$j$	$k$
1	1	$i$	$j$	$k$
$i$	$i$	$-1$	$k$	$-j$
$j$	$j$	$-k$	$-1$	$i$
$k$	$k$	$j$	$-i$	$-1$

The quaternions $\pm 1$, $\pm i$, $\pm j$, and $\pm k$ form a non-Abelian Group of order eight (with multiplication as the group operation) known as $Q_8$.

The quaternions can be written in the form

$$a=a_1+a_2i+a_3j+a_4k.$$ (19)

The conjugate quaternion is given by

$$a^*=a_1-a_2i-a_3j-a_4k.$$ (20)

The sum of two quaternions is then

$$a+b=(a_1+b_1)+(a_2+b_2)i+(a_3+b_3)j+(a_4+b_4)k,$$ (21)

and the product of two quaternions is

$$ab = (a_1b_1-a_2b_2-a_3b_3-a_4b_4)$$

$$\phantom{=} +(a_1b_2+a_2b_1+a_3b_4-a_4b_3)i$$

$$\phantom{=} +(a_1b_3-a_2b_4+a_3b_1+a_4b_2)j$$

$$\phantom{=} +(a_1b_4+a_2b_3-a_3b_2+a_4b_1)k,$$ (22)

so the norm is

$$n(a)=\sqrt{aa^*}=\sqrt{a^*a}=\sqrt{{a_1}^2+{a_2}^2+{a_3}^2+{a_4}^2}\,.$$ (23)

In this notation, the quaternions are closely related to Four-Vectors.

Quaternions can be interpreted as a Scalar plus a Vector by writing

$$a=a_1+a_2i+a_3j+a_4k=(a_1, {\bf a}),$$ (24)

where ${\bf a}\equiv [a_2\,a_3\,a_4]$. In this notation, quaternion multiplication has the particularly simple form

$$q_1q_2=(s_1,{\bf v}_1)\cdot(s_2,{\bf v}_2) =(s_1s_2-{\bf v}_... ...bf v}_2, s_1{\bf v}_2+s_2{\bf v}_1+{\bf v}_1\times {\bf v}_2).$$ (25)

Division is uniquely defined (except by zero), so quaternions form a Division Algebra. The inverse of a quaternion is given by

$$a^{-1}={a^*\over aa^*},$$ (26)

and the norm is multiplicative

$$n(ab)=n(a)n(b).$$ (27)

In fact, the product of two quaternion norms immediately gives the Euler Four-Square Identity.

A rotation about the Unit Vector $\hat {\bf n}$ by an angle $\theta$ can be computed using the quaternion

$$q=(s,{\bf v})=(\cos({\textstyle{1\over 2}}\theta),\hat {\bf n}\sin({\textstyle{1\over 2}}\theta))$$ (28)

(Arvo 1994, Hearn and Baker 1996). The components of this quaternion are called Euler Parameters. After rotation, a point $p=(0,{\bf p})$ is then given by

$$p'=qpq^{-1}=qpq^*,$$ (29)

since $n(q)=1$. A concatenation of two rotations, first $q_1$ and then $q_2$, can be computed using the identity

$$q_2(q_1pq_1^*)q_2^*=(q_2q_1)p(q_1^*q_2^*)=(q_2q_1)p(q_2q_1)^*$$ (30)

(Goldstein 1980).

See also Biquaternion, Cayley-Klein Parameters, Complex Number, Division Algebra, Euler Parameters, Four-Vector, Octonion

References

Quaternions

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Hamilton, W. R. The Mathematical Papers of Sir William Rowan Hamilton. Cambridge, England: Cambridge University Press, 1967.

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Joly, C. J. A Manual of Quaternions. London: Macmillan, 1905.

Julstrom, B. A. ``Using Real Quaternions to Represent Rotations in Three Dimensions.'' UMAP Modules in Undergraduate Mathematics and Its Applications, Module 652. Lexington, MA: COMAP, Inc., 1992.

Kelland, P. and Tait, P. G. Introduction to Quaternions, 3rd ed. London: Macmillan, 1904.

Nicholson, W. K. Introduction to Abstract Algebra. Boston, MA: PWS-Kent, 1993.

Tait, P. G. An Elementary Treatise on Quaternions, 3rd ed., enl. Cambridge, England: Cambridge University Press, 1890.

Tait, P. G. ``Quaternions.'' Encyclopædia Britannica, 9th ed. ca. 1886. ftp://ftp.netcom.com/pub/hb/hbaker/quaternion/tait/Encyc-Brit.ps.gz.