= Complex matrix representation of quaternions
{title2=$\begin{pmatrix}a+ib&c+id\\-c+id&a-ib\end{pmatrix}$}
The displayed complex <matrices>, with real coefficients, form a four-dimensional real <associative algebra>. Their three imaginary <basis> <matrices> square to minus the identity and satisfy $JK=L$, $KL=J$, $LJ=K$, with the reversed products negative. For $\alpha=aI+bJ+cK+dL$, quaternionic conjugation gives $\alpha\overline\alpha=(a^2+b^2+c^2+d^2)I$. Every nonzero <matrix> therefore has inverse $\overline\alpha/(a^2+b^2+c^2+d^2)$ inside this algebra. This realizes the real <division algebra> of <quaternions> within $M_2(\mathbb C)$; it is not a complex <vector> subspace.
Back to article page
