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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],
\end{displaymath} (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}}
\end{displaymath} (2)

of the four matrices
$\displaystyle {\hbox{\sf U}}$ $\textstyle \equiv$ $\displaystyle \left[\begin{array}{cc}1 & 0\\  0 & 1\end{array}\right]$ (3)
$\displaystyle {\hbox{\sf I}}$ $\textstyle \equiv$ $\displaystyle \left[\begin{array}{cc}i & 0\\  0 & -i\end{array}\right]$ (4)
$\displaystyle {\hbox{\sf J}}$ $\textstyle \equiv$ $\displaystyle \left[\begin{array}{cc}0 & 1\\  -1 & 0\end{array}\right]$ (5)
$\displaystyle {\hbox{\sf K}}$ $\textstyle \equiv$ $\displaystyle \left[\begin{array}{cc}0 & i\\  i & 0\end{array}\right].$ (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
$\displaystyle {\hbox{\sf I}}^2$ $\textstyle =$ $\displaystyle -{\hbox{\sf U}}$ (7)
$\displaystyle {\hbox{\sf J}}^2$ $\textstyle =$ $\displaystyle -{\hbox{\sf U}}$ (8)
$\displaystyle {\hbox{\sf K}}^2$ $\textstyle =$ $\displaystyle -{\hbox{\sf U}}.$ (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}},
\end{displaymath} (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

$\displaystyle i$ $\textstyle \equiv$ $\displaystyle \left[\begin{array}{cccc}0 & 1 & 0 & 0 \\  -1 & 0 & 0 & 0 \\  0 & 0 & 0 & 1 \\  0 & 0 & -1 & 0\end{array}\right]$ (11)
$\displaystyle j$ $\textstyle \equiv$ $\displaystyle \left[\begin{array}{cccc}0 & 0 & 0 & -1 \\  0 & 0 & -1 & 0 \\  0 & 1 & 0 & 0 \\  1 & 0 & 0 & 0\end{array}\right]$ (12)
$\displaystyle k$ $\textstyle \equiv$ $\displaystyle \left[\begin{array}{cccc}0 & 0 & -1 & 0 \\  0 & 0 & 0 & 1 \\  1 & 0 & 0 & 0 \\  0 & -1 & 0 & 0\end{array}\right]$ (13)
$\displaystyle 1$ $\textstyle \equiv$ $\displaystyle \left[\begin{array}{cccc}1 & 0 & 0 & 0 \\  0 & 1 & 0 & 0 \\  0 & 0 & 1 & 0 \\  0 & 0 & 0 & 1\end{array}\right].$ (14)

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

i^2 = j^2 = k^2 = -1
\end{displaymath} (15)

ij = -ji = k
\end{displaymath} (16)

jk = -kj = i
\end{displaymath} (17)

ki = -ik = j.
\end{displaymath} (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

\end{displaymath} (19)

The conjugate quaternion is given by
\end{displaymath} (20)

The sum of two quaternions is then
\end{displaymath} (21)

and the product of two quaternions is
$\displaystyle ab$ $\textstyle =$ $\displaystyle (a_1b_1-a_2b_2-a_3b_3-a_4b_4)$  
  $\textstyle \phantom{=}$ $\displaystyle +(a_1b_2+a_2b_1+a_3b_4-a_4b_3)i$  
  $\textstyle \phantom{=}$ $\displaystyle +(a_1b_3-a_2b_4+a_3b_1+a_4b_2)j$  
  $\textstyle \phantom{=}$ $\displaystyle +(a_1b_4+a_2b_3-a_3b_2+a_4b_1)k,$ (22)

so the norm is
\end{displaymath} (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}),
\end{displaymath} (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}_... v}_2, s_1{\bf v}_2+s_2{\bf v}_1+{\bf v}_1\times {\bf v}_2).
\end{displaymath} (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^*},
\end{displaymath} (26)

and the norm is multiplicative
\end{displaymath} (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))
\end{displaymath} (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
\end{displaymath} (29)

since $n(q)=1$. A concatenation of two rotations, first $q_1$ and then $q_2$, can be computed using the identity
\end{displaymath} (30)

(Goldstein 1980).

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



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Arvo, J. Graphics Gems II. New York: Academic Press, pp. 351-354 and 377-380, 1994.

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© 1996-9 Eric W. Weisstein