Abstract Algebra/Quaternions

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The set of Quaternions is an algebraic structure first studied by the Irish mathematician William Rowan Hamilton, in the hopes of constructing a generalization for complex numbers. When first discovered, quaternions generated a lot of excitement among mathematicians and physicists alike, for it was hoped that quaternions would provide a "unified theory" of mechanics and electromagnetism. Although these hopes proved to be unfounded, quaternions are still considered interesting as well as useful mathematical entities.

Definition[edit]

A Quaternion is an ordered 4-tuple q=(a,b,c,d), where a,b,c,d\in\mathbb{R}. A quaternion is often denoted as q=a+bi+cj+dk (Observe the analogy with complex numbers). The set of all quaternions is denoted by \mathbb{H}.

It is straightforward to define component-wise addition and scalar multiplication on \mathbb{H}, making it a real vector space.

The rule for multiplication was a product of Hamilton's ingenuity. He discovered what are known as the Bridge-stone Equations:

i^2 = j^2 = k^2 = ijk = -1

From the above equations alone, it is possible to derive rules for the pairwise multiplication of i, j, and k:

 ij=k,  jk=i,  ki=j (positive cyclic permutations)

ji=-k,  kj=-i,  ik=-j (negative cyclic permutations).

Using these, it is easy to define a general rule for multiplication of quaternions. Because quaternion multiplication is not commutative, \mathbb{H} is not a field. However, every nonzero quaternion has a multiplicative inverse (see below), so the quaternions are an example of a non-commutative division ring. It is important to note that the non-commutative nature of quaternion multiplication makes it impossible to define the quotient p/q of two quaternions p and q unambiguously, as the quantities pq^{-1} and q^{-1}p are generally different.

Like the more familiar complex numbers, the quaternions have a conjugation, often denoted by a superscript star: q^*. The conjugate of the quaternion q=a+bi+cj+dk is q^*=a-bi-cj-dk. As is the case for the complex numbers, the product qq^* is always a positive real number equal to the sum of the squares of the quaternion's components. Using this fact, it is fairly easy to show that the multiplicative inverse of a general quaternion q is given by

q^{-1}=\frac{q^*}{qq^*}

where division is defined since qq^* is a scalar. Note that, unlike in the complex case, the conjugate q^* of a quaternion q can be written as a polynomial in q:

q^*=-\frac{1}{2}(q+iqi+jqj+kqk).

The quaternions are isomorphic to the Clifford algebra C2(R) and the even subalgebra of C3(R).

Pauli Spin Matrices[edit]

Quaternions are closely related to the Pauli spin matrices of Quantum Mechanics. The Pauli matrices are often denoted as
\sigma_1=\begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix} , \sigma_2=\begin{pmatrix}0 & -i \\ i & 0\end{pmatrix} , \sigma_3=\begin{pmatrix}1 & 0 \\ 0 & -1\end{pmatrix}

(Where i is the well known quantity \sqrt{-1} of complex numbers)

The 2×2 identity matrix is sometimes taken as \sigma_0. It can be shown that S, the real linear span of the matrices \sigma_0, i\sigma_1, i\sigma_2 and i\sigma_3, is isomorphic to the set of all quaternions, \mathbb{H}. For example, take the matrix product below:

\begin{pmatrix}i & 0 \\ 0 & -i\end{pmatrix} \begin{pmatrix}0 & 1 \\ -1 & 0\end{pmatrix} = \begin{pmatrix}0 & i \\ i & 0\end{pmatrix}

Or, equivalently:

i\sigma_3 i\sigma_2 = i\sigma_1

All three of these matrices square to the negative of the identity matrix. If we take  1=\sigma_0, i=i\sigma_3, j=i\sigma_2, and k=i\sigma_1, it is easy to see that the span of the these four matrices is "the same as" (that is, isomorphic to) the set of quaternions \mathbb{H}.

Exercise[edit]

  1. Using the Bridge-stone equations, explicitly state the rule of multiplication for general quaternions, that is, given q_1=a_1+b_1i+c_1j+d_1k and q_2=a_2+b_2i+c_2j+d_2k, give the components of their product q=q_1q_2

Reference[edit]

  • E.T. Bell, Men of Mathematics, Simon & Schuster, Inc.
  • The Wikipedia article on Pauli Spin Matrices