M_3(H) matrix algebra

$M_{3}(\mathbb H)$ consists of $3$ -by- $3$ matrices with entries in H, with entrywise addition and scalar multiplication and product

(XY)_{ij}=\sum_{r=1}^{3}X_{ir}Y_{rj}.

Its unit is $I_{3}$ . The scalar field for this catalogue object is $\mathbb R$ .

Dimension and multiplication

The real dimension is $36$ , since there are $9$ 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 $X(YZ)$ . The standard multiplication is real-bilinear. Choosing a copy of $\mathbb C\subset\mathbb H$ does not make it a complex algebra: those scalars need not commute with matrix entries.

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.