For a natural number, the special unitary group is the group of isometries of the -dimensional complex Hilbert space which preserve the volume form on this space. It is the subgroup of the unitary group consisting of the unitary matrices with determinant .
More generally, for any complex vector space equipped with a nondegenerate Hermitian form? , is the group of isometries of which preserve the volume form derived from . One may write if is obvious, so that is the same as . By , we mean , where has positive eigenvalue?s and negative ones.