1.1.5 Orthogonal Groups

The or­thog­o­nal group is a sub­group of $GL(n,F)$, with in­vert­ible linear $n \times n$ ma­tri­ces over the field F, with degree $n$, and is de­fined by $M \bullet M^{*} = I$, for $M^{*}$, the con­ju­gate trans­pose of the matrix $M$. The group and its defin­ing prop­er­ties are

where $M^{*}$ is equiv­a­lent in this case to the matrix trans­pose$\ M^{T}$, for $M\in \mathbb{R}$. Dif­fer­ent field values for F exist, amongst which, we find the real num­bers $\mathbb{R}$, the com­plex num­bers $\mathbb{C}$, and the quater­nion num­bers $\mathbb{H}$. Based on the field values, spe­cif­ic de­f­i­n­i­tions for the or­thog­o­nal groups arise:

• $\mathcal{O}\left( n,\mathbb{R} \right)$ is de­not­ed $O(n)$ and is called the or­thog­o­nal group,

• $\mathcal{O}\left( n,\mathbb{C} \right)$ is de­not­ed $U(n)$ and is called the uni­tary group,

• $\mathcal{O}\left( n,\mathbb{H} \right)$ is de­not­ed $Sp(n)$ and is called the sym­plec­tic group.