M_2(H_s) matrix algebra

$M_{2}(\mathbb H_s)$ consists of $2$ -by- $2$ matrices with entries in H_s, with entrywise addition and scalar multiplication and product

(XY)_{ij}=\sum_{r=1}^{2}X_{ir}Y_{rj}.

Its unit is $I_{2}$ . The scalar field for this catalogue object is $\mathbb R$ .

Dimension and multiplication

The real dimension is $16$ , since there are $4$ entries with $4$ real coordinates each.

Matrix multiplication is associative because the coefficient algebra is associative; distributivity and rebracketing justify the equality of each entry of $(XY)Z$ and $X(YZ)$ . Replacing each split quaternion by its two-by-two real matrix identifies this algebra with $M_{4}(\mathbb R)$ .

Size convention

For every size at least two, $E_{11}E_{22}=0$ exhibits nonzero zero divisors. Matrix size is not the dimension of the whole algebra.