Definition 41.11.1 (Matrix Order).label Let $S$ be a complex vector space equipped with a complex conjugation $*: S \to S$, then $S$ is matrix ordered if:

1. (1)

   For each $n \in \natp$, $M_{n}(S)_{sa}$ is equipped with a vector space ordering.

2. (2)

   For each $m, n \in \natp$, $A \in M_{n \times m}(\complex)$, and $x \in M_{n}(S)_{sa}$ with $x \ge 0$, $A^{*}xA \ge 0$ in $M_{m}(S)$.