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Hamiltonian Quaternion \(\Bbb H\) is a four dimensional vector space over \(\Bbb R\) with the basis \(\{1,i,j,k\}\), and its multiplication is characterized by
\[\begin{align*} i^2=j^2=k^2=-1, \\ ij=-ji=k, \\ jk=-kj=i, \\ ki=-ik=j. \end{align*}\]有乘法的 vector space 叫做 algebra！
Let \(V\) be the subspace of \(\Bbb H\) spanned by \({i,j,k}\). Let \(\Bbb R\) be the subspace of \(Bbb H\) spanned by \(1\). Then we have
\[\Bbb H = \Bbb R \oplus V\]\(\oplus\) 是 direct sum！
That is to say, every element of \(\Bbb H\) can be written as \(a+\vec u\), where \(a\in \Bbb R\) and \(\vec u \in V\).
For \(\vec u, \vec v \in V\), we have
\[\vec u\vec v = -\vec u\cdot\vec v + \vec u\times \vec v.\]
Proof
展開即證明。
Let \(T\) be a counter-clockwise rotation of \(\theta\)-degree with respect to the \(\vec u\)-axis, where \(\vec u\) is an unit vector in \(\Bbb{R}^3\). Consider the quaternion
\[q=\cos{\theta\over 2} + \sin{\theta\over 2}\vec u.\]Then for all \(\vec v \in \Bbb R\),
\[T(\vec v) = q\vec v \bar q.\]
\(q\bar q = 1\).
Proof
Let \(T_q(z) = qz\bar q\) be a map from \(\Bbb H\) to \(\Bbb H\). We can see that \(T_q\) is a linear map.
\[\begin{align*} T_q(x+y) &= q(x+y)\bar{q} = qx\bar{q} + qy\bar{q}, \\ T_q(ax) &= q(ax\bar{q}) = a(qx\bar{q}). \end{align*}\]
We shall show that \(T_q(V)\subset V\) and \(T_q\vert_V = T\).
Note that \(\vec u,q,\bar q\) all commute. Thus \(T_q(\vec u) = \vec uq\bar q = \vec u = T(\vec u)\). Now choose \(\vec v\in V\), such that \(\vec v\cdot \vec u = 0\) and whise norm is \(1\). Then \(\{\vec u, \vec v, \vec u\times \vec v \}\) forms an orthonormal basis of \(V\). It remains to show that \(T_q(\vec v) = T(\vec v)\) and \(T_q(\vec u\times \vec v) = T(\vec u\times \vec v)\).
轉 basis 中的 vector 就夠了！
First, note that
\[T(v) = \cos\theta\vec v + \sin\theta(\vec u\times \vec v).\]用平面來想：\(\vec v\) 是 \(x\) 軸，\((\vec u\times \vec v)\) 是 \(y\) 軸。
Next, we have
\[T_q(\vec v) = q\vec v \vec q = \cdots = T(\vec v).\]Replacing \(\vec v\) by \(\vec u\times \vec v\) leads to a similar result. ◼
比起用矩陣來表示旋轉，用四元數簡潔方便多了！
See rings and fields.
Let \(\phi:R\to R'\) be a ring homomorphism with kernel \(K\). Then the additive cosets of \(H\) form a ring \(R/K\), with addition and multiplication given by
\[\begin{align*} +:(a+H, b+H) \mapsto (a+b)+H \\ \times: (a+H, b+H) \mapsto (ab) + H. \end{align*}\]
Proof
We need to prove that
The kernel \(H\) above satisfies
\[ah, hb \in H,\]for all \(a,b \in R\) and \(h\in H\).
用來輔助證明乘法是 well-defined 的。
We have
\[\begin{align*} \Bbb H &= \Bbb R \oplus \Bbb Ri \oplus \Bbb Rj \oplus \Bbb Rk \\ &= \Bbb C \oplus \Bbb Cj. \end{align*}\]Then, for any \(\alpha\in \Bbb H, \alpha = \alpha_1 + \alpha_2j\), we have the following homomorphism:
\[\begin{align*} \rho: \Bbb H&\to \text{M}_2(\Bbb C), \\ \alpha &\mapsto [L_\alpha]_\beta, \end{align*}\]where \(\beta = \{1, j\}\).
Check Matrix representation.