Quaternions Subring of Complex Matrix Space
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Theorem
The Ring of Quaternions is a subring of the matrix space $\map {\MM_\C} 2$.
Proof
From Matrix Form of Quaternion it is clear that the quaternions $\H$ can be expressed in matrix form, as elements of $\map {\MM_\C} 2$.
Thus $\H \subseteq \map {\MM_\C} 2$.
As the quaternions form a ring, the result follows by definition of subring.
$\blacksquare$
Sources
- 1970: B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra ... (previous) ... (next): $\S 2.1$: Subrings: Examples $2$