Herm_n(C) Jordan algebra

For integers $n\geq1$ , The Herm_n(C) Jordan algebra is the $\mathbb R$ -vector space

\operatorname{Herm}_{n}(\mathbb C)=\{X\in M_{n}(\mathbb C):X^*=X\}, \qquad X\circ Y=\tfrac12(XY+YX),

with unit $I_{n}$ . Here $X^*=\overline X^T$ , using the standard conjugation of the coefficient algebra.

Coordinates and dimension

There are $n$ diagonal scalar coordinates and $n(n-1)/2$ independent coefficient-algebra entries. Thus the dimension over $\mathbb R$ is $n+2n(n-1)/2$ . In particular, the usual Hermitian complex matrices form a real vector space; multiplication by $i$ does not preserve self-adjointness.

Jordan identity and range

The fixed subspace is closed under $X\circ Y$ , because $(XY)^*=Y^*X^*$ . Associativity of the ambient matrix algebra proves the Jordan identity, so the construction works in every positive size.