M_n(S) matrix algebra

For each integer $n\geq1$ , $M_{n}(\mathbb S)$ consists of $n$ -by- $n$ matrices with entries in S, with entrywise addition and scalar multiplication and product

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

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

Dimension and multiplication

Each of the $n^2$ entries contributes $16$ real coordinates, so the real dimension is $16n^2$ .

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

The entries for sizes one, two, and three are separate catalogue objects; this record is the parameterized family. Matrix size is not the dimension of the algebra.