Quaternions Defined by Matrices

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Theorem

Let $\mathbf 1, \mathbf i, \mathbf j, \mathbf k$ denote the following four elements of the matrix space $\map {\MM_\C} 2$:

$\mathbf 1 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}

\qquad \mathbf i = \begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix} \qquad \mathbf j = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} \qquad \mathbf k = \begin{bmatrix} 0 & i \\ i & 0 \end{bmatrix} $ where $\C$ is the set of complex numbers.


Then $\mathbf 1, \mathbf i, \mathbf j, \mathbf k$ are related to each other in the following way:

\(\ds \mathbf i \mathbf j = - \mathbf j \mathbf i\) \(=\) \(\ds \mathbf k\)
\(\ds \mathbf j \mathbf k = - \mathbf k \mathbf j\) \(=\) \(\ds \mathbf i\)
\(\ds \mathbf k \mathbf i = - \mathbf i \mathbf k\) \(=\) \(\ds \mathbf j\)
\(\ds \mathbf i^2 = \mathbf j^2 = \mathbf k^2 = \mathbf i \mathbf j \mathbf k\) \(=\) \(\ds -\mathbf 1\)


Proof

This is demonstrated by straightforward application of conventional matrix multiplication:


\(\ds \mathbf i \mathbf j\) \(=\) \(\ds \begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix} \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}\)
\(\ds \) \(=\) \(\ds \begin{bmatrix} i \cdot 0 + 0 \cdot -1 & i \cdot 1 + 0 \cdot 0 \\ 0 \cdot 0 + -i \cdot -1 & 0 \cdot 1 + -i \cdot 0 \end{bmatrix}\)
\(\ds \) \(=\) \(\ds \begin{bmatrix} 0 & i \\ i & 0 \end{bmatrix}\)
\(\ds \) \(=\) \(\ds \mathbf k\)


\(\ds -\mathbf j \mathbf i\) \(=\) \(\ds -\begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} \begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix}\)
\(\ds \) \(=\) \(\ds -\begin{bmatrix} 0 \cdot i + 1 \cdot 0 & 0 \cdot 0 + 1 \cdot -i \\ -1 \cdot i + 0 \cdot 0 & -1 \cdot 0 + 0 \cdot -i \end{bmatrix}\)
\(\ds \) \(=\) \(\ds -\begin{bmatrix} 0 & -i \\ -i & 0 \end{bmatrix}\)
\(\ds \) \(=\) \(\ds \begin{bmatrix} 0 & i \\ i & 0 \end{bmatrix}\)
\(\ds \) \(=\) \(\ds \mathbf k\)



\(\ds \mathbf j \mathbf k\) \(=\) \(\ds \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} \begin{bmatrix} 0 & i \\ i & 0 \end{bmatrix}\)
\(\ds \) \(=\) \(\ds \begin{bmatrix} 0 \cdot 0 + 1 \cdot i & 0 \cdot i + 1 \cdot 0 \\ -1 \cdot 0 + 0 \cdot i & -1 \cdot i + 0 \cdot 0 \end{bmatrix}\)
\(\ds \) \(=\) \(\ds \begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix}\)
\(\ds \) \(=\) \(\ds \mathbf i\)


\(\ds -\mathbf k \mathbf j\) \(=\) \(\ds -\begin{bmatrix} 0 & i \\ i & 0 \end{bmatrix} \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}\)
\(\ds \) \(=\) \(\ds -\begin{bmatrix} 0 \cdot 0 + i \cdot -1 & 0 \cdot 1 + i \cdot 0 \\ i \cdot 0 + 0 \cdot -1 & i \cdot 1 + 0 \cdot 0 \end{bmatrix}\)
\(\ds \) \(=\) \(\ds -\begin{bmatrix} -i & 0 \\ 0 & i \end{bmatrix}\)
\(\ds \) \(=\) \(\ds \begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix}\)
\(\ds \) \(=\) \(\ds \mathbf i\)



\(\ds \mathbf k \mathbf i\) \(=\) \(\ds \begin{bmatrix} 0 & i \\ i & 0 \end{bmatrix} \begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix}\)
\(\ds \) \(=\) \(\ds \begin{bmatrix} 0 \cdot i + i \cdot 0 & 0 \cdot 0 + i \cdot -i \\ i \cdot i + 0 \cdot 0 & i \cdot 0 + 0 \cdot -i \end{bmatrix}\)
\(\ds \) \(=\) \(\ds \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}\)
\(\ds \) \(=\) \(\ds \mathbf j\)


\(\ds -\mathbf i \mathbf k\) \(=\) \(\ds -\begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix} \begin{bmatrix} 0 & i \\ i & 0 \end{bmatrix}\)
\(\ds \) \(=\) \(\ds -\begin{bmatrix} i \cdot 0 + 0 \cdot i & i \cdot i + 0 \cdot 0 \\ 0 \cdot 0 + -i \cdot i & 0 \cdot i + -i \cdot 0 \end{bmatrix}\)
\(\ds \) \(=\) \(\ds -\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}\)
\(\ds \) \(=\) \(\ds \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}\)
\(\ds \) \(=\) \(\ds \mathbf j\)



\(\ds \mathbf i^2\) \(=\) \(\ds \begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix} \begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix}\)
\(\ds \) \(=\) \(\ds \begin{bmatrix} i \cdot i + 0 \cdot 0 & i \cdot 0 + 0 \cdot -i \\ 0 \cdot i + -i \cdot 0 & -i \cdot 0 + -i \cdot -i \end{bmatrix}\)
\(\ds \) \(=\) \(\ds \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix}\)
\(\ds \) \(=\) \(\ds -\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\)
\(\ds \) \(=\) \(\ds -\mathbf 1\)


\(\ds \mathbf j^2\) \(=\) \(\ds \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}\)
\(\ds \) \(=\) \(\ds \begin{bmatrix} 0 \cdot 0 + 1 \cdot -1 & 0 \cdot 1 + 1 \cdot 0 \\ -1 \cdot 0 + 0 \cdot -1 & -1 \cdot 1 + 0 \cdot 0 \end{bmatrix}\)
\(\ds \) \(=\) \(\ds \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix}\)
\(\ds \) \(=\) \(\ds -\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\)
\(\ds \) \(=\) \(\ds -\mathbf 1\)


\(\ds \mathbf k^2\) \(=\) \(\ds \begin{bmatrix} 0 & i \\ i & 0 \end{bmatrix} \begin{bmatrix} 0 & i \\ i & 0 \end{bmatrix}\)
\(\ds \) \(=\) \(\ds \begin{bmatrix} 0 \cdot 0 + i \cdot i & 0 \cdot i + i \cdot 0 \\ i \cdot 0 + 0 \cdot i & i \cdot i + 0 \cdot 0 \end{bmatrix}\)
\(\ds \) \(=\) \(\ds \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix}\)
\(\ds \) \(=\) \(\ds -\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\)
\(\ds \) \(=\) \(\ds -\mathbf 1\)



\(\ds \mathbf i \mathbf j \mathbf k\) \(=\) \(\ds \paren {\mathbf i \mathbf j} \mathbf k\) Matrix Multiplication is Associative
\(\ds \) \(=\) \(\ds \mathbf k \mathbf k\) from above: $\mathbf i \mathbf j = \mathbf k$
\(\ds \) \(=\) \(\ds -\mathbf 1\) from above: $\mathbf k^2 = - \mathbf 1$


$\blacksquare$


Sources

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