M_1(S) matrix algebra

$M_{1}(\mathbb S)$ consists of $1$ -by- $1$ matrices with entries in S, 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 $16$ , since there are $1$ entries with $16$ real coordinates each.

Two-factor matrix products are unambiguous, but multiplication is not associative: the coefficient algebra embeds in a diagonal corner, so its nonzero associators survive. No associative-algebra or ring view is assigned. Symmetrizing this product must not be assumed to give a Jordan algebra.

Size convention

For size one, $[a]\mapsto a$ identifies this named matrix construction with the coefficient algebra, with the same product and scalar field.