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.

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[edit] Definition

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). We denote the set of all quaternions is denoted by H.

Analoguosly, we can define component-wise addition on H.

The rule for multiplication( * ) was Hamilton's ingenuinity, he first came up with what are known as the Bridge-stone Equations:
i2 = j2 = k2 = − 1

i * j = k,j * k = i,k * i = j (positive cyclic permutation)

j * i = − k,k * j = − i,i * k = − j (negative cyclic permutation)

Using these, it is easy to define a general rule for multiplication of quaternions. Unlike for complex numbers, we cannot meaningfully define a multiplicative "inverse" for a quaternion. Hence, division by quaternions is meaningless, implying that H is not a field, but a ring.

[edit] Pauli Spin Matrices

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 \sqrt{-1} of complex numbers)

It can be shown that S, the real linear span of the Pauli matrices, is isomorphic to the set of all quaternions, H.

[edit] Exercise

  1. Using the Bridge-stone equations, explicitly state the rule of multiplication for general quaternions, that is, given q1 = a1 + b1i + c1j + d1k and q2 = a2 + b2i + c2j + d2k, give the components of their product q = q1 * q2

[edit] Reference

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