S orthogonal matrices and S symmetries

Let S∈Mn(C)S∈Mn(C) be nonsingular. A Q∈Mn(C)Q∈Mn(C) is called S orthogonal   if QTSQ=SQTSQ=S. An S orthogonal H is called an S symmetry   if rank(H−I)=1rank(H−I)=1. We give conditions on S so that every S orthogonal can be written as a product of S symmetries. We give conditions on S so that some S orthogonal cannot be written as a product of S symmetries. Moreover, we determine conditions on S so that there are no S symmetries.