Rational matrix algebra M_2(Q)

The rational matrix algebra $M_2(\mathbb Q)$ is the vector space of $2\times 2$ matrices with rational entries, with entrywise addition and scalar multiplication, and the associative product

(AB)_{ij}=\sum_{r=1}^{2} A_{ir}B_{rj}.

Its multiplicative unit is the identity matrix $I_2$ .

Basis and dimensions

The matrix units $E_{ij}$ form a rational basis of size $2^2$ , with $E_{ij}E_{rs}=\delta_{jr}E_{is}$ . Thus its rational dimension is 4. The center consists of scalar matrices.

Endomorphism interpretation

After choosing a basis of $\mathbb Q^2$ , this algebra identifies with $\operatorname{End}_{\mathbb Q}(\mathbb Q^2)$ , with matrix product corresponding to composition. Its units form $\mathrm{GL}_2(\mathbb Q)$ . These are units of this algebra; an automorphism of the matrix algebra is a different sort of map.

Low-dimensional boundary

For size one, scalar matrices give $M_1(\mathbb Q)\cong\mathbb Q$ as unital rational algebras. For every size at least two, $E_{11}$ and $E_{22}$ are nonzero with zero product, so the matrix algebra is not a division algebra.

Quadratic-field example

Using the basis $1,i$ identifies the rational-linear endomorphisms of $\mathbb Q(i)$ with this matrix algebra. Multiplication by $a+bi$ corresponds to $\left(\begin{smallmatrix}a&-b\\b&a\end{smallmatrix}\right)$ , a two-dimensional subalgebra of the four-dimensional matrix algebra.