M_1(H) matrix algebra

$M_{1}(\mathbb H)$ consists of $1$ -by- $1$ matrices with entries in H, with entrywise addition and scalar multiplication and product

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

Its unit is $I_{1}$ . The scalar field for this catalogue object is $\mathbb R$ .

Dimension and multiplication

The real dimension is $4$ , since there are $1$ 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 size one, $[a]\mapsto a$ identifies this named matrix construction with the coefficient algebra, with the same product and scalar field.